lE. 


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ncs. 


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onal  ap- 

t  unique, 
f  calling 


isses  of 


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SHAW'S   NEW   SERIES 

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A 

u:niyersity 


ALGEBRA 


COMPRISING 


I. -A  COMPENDIOUS,  YET  COMPLETE  AND  THOROUGH  COURSE  IN  ELEMENTARY 
ALGEBRA,  AND 

II.— AN   ADVANCED    COURSE    IN  ALGEBRA,  suppiciently   extendbd   to    meet  tub 

WAMT3   or  OUB  UKirEBSITlES,  COLLEGES,   AND  SCHOOLS  Of  SCIENCE. 


EDWARD   pLNEY, 

PROVE880R  OP   MATHBMATICS  IN  THK  UNITERSITY  OP  MICHIQAN,   AND  AUTHOR  OV 
A  SERIES  OF  MATHEMATICS. 


NEW   YORK: 

SHELDON    &    COMPANY, 

No.  8    MURRAY   STREET. 
1885. 


oj-e     _?. 


PROF.  OLNEY'S  MATHEMATICAL  COURSE. 


INTRODUCTION  TO  ALGEBRA 

COMPLETE  ALGEBRA        

KEY  TO  COMPLETE  ALGEBRA  --.--- 

UNIVERSITY  ALGEBRA 

KEY"  TO  UNIVERSITY  ALGEBRA 

A  VOLUME  OF  TEST  EXAMPLES  IN  ALGEBRA 

ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY 

ELEMENTS  OF  GEOMETRY  AND  TRIGONOMETRY,  University 
Edition 

ELEMENTS  OF  GEOMETRY,  separate 

ELEMENTS  OF  TRIGONOMETRY,  separate 

GENERAL  GEOMETRY  AND  CALCULUS    

BELLOWS'  TRIGONOMETRY 

*  ^  »  m  * 

PROF.  OLNEY^S  SERIES  OF  ARITHMETICS. 

PRIMARY  ARITHMETIC 

ELEMENTS  OF  ARITHMETIC  -------- 

PRACTICAL  ARITHxMETIC 

SCIENCE  OF  ARITHMETIC       -  ^ 


Entered  according  to  Act  of  Congress,  in  the  year  1873, 

By  SHELDON  &  COMPANY, 

In  the  Office  of  the  Librarian  of  Congress,  at  Washington. 

UN.  A. 


The  Author's  Complete  School  Algebra  was  written  to  meet  the  wants 
of  our  Common  and  High  Schools  and  Academies,  and  to  afford  adequate  prepara- 
tion for  entering  our  best  Colleges,  Schools  of  Science,  and  Universities. 

The  present  volume  is  designed  for  use  in  these  advanced  courses  of  training. 
Thus,  while  it  is  thought  that  the  former  affords  as  extended  a  course  in  Algebra 
as  is  expedient  for  the  preparatory  schools,  it  is  believed  that  this  will  be  found 
to  contain  all  that  these  higher  schools  require. 

It  was  deemed  necessary  to  make  the  work  a  complete  treatise,  including  the 
Elements,  for  purposes  of  reference,  and  for  reviews,  and  also  in  consideration 
of  the  fact  that  our  higher  institutions  have  various  standards  of  requirement 
for  admission.  In  fact,  there  are  few  students  of  Higher  Algebra  who  do  not 
find  it  necessary  to  have  the  Elements  at  hand  for  occasional  consultation. 

This  Elementary  portion  is  embraced  in  the  first  150  pages,  and  contains  all 
the  definitions,  principles,  rules,  and  demonstrations  of  the  Complete  School 
Algebra,  with  an  abundant  collection  of  Neio  Examines  ;  but  from  it  all  ele- 
mentary illustrations,  explanations,  solutions,  and  suggestions,  are  omitted. 
The  whole  is  so  arranged  as  to  secure  readiness  of  reference  and  convenience 
of  review  by  somewhat  mature  students. 

The  subjects  treated  in  Part  III.,  which  constitutes  the  Advanced  Course 
proper,  will  be  best  seen  by  turning  to  the  Table  of  Contents.  In  this  place 
the  author  wishes  merely  to  call  attention  to  a  few  of  the  distinguishing  fea- 
tures of  this  Part. 

1,  The  conception  of  Function  and  Variable  is  introduced  at  once,  and  is 
made  familiar  by  such  use  of  it  as  mathematicians  are  constantly  making.  No 
one  needs  to  be  told  that  this  conception  lies  at  the  foundation  of  all  higher 
algebraic  discussion  ;  yet,  strangely  enough,  the  very  terms  are  scarcely  to  be 
found  in  our  common  text-books,  and  the  practical  use  of  the  conception  is 
totally  wanting.  80054S 


iV  PREFACE. 

2.  The  first  chapter  in  the  Advanced  Course  is  given  to  an  olcmpntary  and 
practical  exposition  of  the  Infinitesimal  Analysis.  Tlie  autlior  knows  from  liis 
own  experience,  and  from  tliat  of  many  others,  that  tliis  subject  presents  no 
peculiar  diflBculties  to  ordinary  minds  ;  and  everybody  knows  tliat  it  is  cmly  by 
this  analysis  that  the  development  of  functions,  as  in  the  Binomial  Formula, 
Logarithmic  Series,  etc.,  the  general  relation  of  function  and  variable,  tlie 
evolution  of  many  of  the  principles  requisite  in  solving  the  Higher  Equations, 
and  many  other  subjects,  are  ever  treated  by  mathematicians,  except  wlien  thoy 
attempt  to  make  Algebras.  No  mathematician  thinks  of  using  the  clumsy 
and  antiquated  processes  by  which  we  have  been  accustomed  to  teach  our  i>upil3 
in  algebra  to  demonstrate  the  Binomial  Formula,  produce  the  Logarithmic  Series, 
deduce  the  law  of  derived  polynomials,  examine  the  relative  rate  of  change  of  a 
function  and  its  variable,  etc.,  except  when  he  is  teaching  the  tyro.  Why  not, 
then,  dismiss  forever  these  processes,  and  let  tin;  pupil  enter  at  once  upon  those 
elegant  and  productive  methods  of  thinking  which  he  will  ever  after  use  ? 

3.  By  the  introduction  of  a  short  chapter  on  Loci  of  Equations,  which  any 
one  can  read  even  without  a  knowledge  of  Elementary  Geometry,  and  which 
in  itself  is  always  interesting  to  the  pupil,  and  of  fundamental  use  in  the  sub- 
sequent course,  all  the  more  abstruse  principles  of  tlie  Tfieory  of  EqwUions  are 
illustrated,  and  the  student  is  thus  enabled  to  see  the  truth,  as  well  as  to  demon- 
strate it  abstractly.     How  great  an  advantage  this  is,  no  experienced  teacher 

to  be  told. 


4.  In  the  treatment  of  the  Higher  E<iuations,  while  some  things  have  been 
discarded  which  everylxxly  knows  to  be  worthless,  but  which  have  in  some 
way  found  a  place  in  our  text-books,  a  far  more  full  and  clear  discussion  of 
practical  principles  and  methods  is  given,  than  is  found  in  any  of  the  trea- 
tises in  common  use. 

5.  The  important  but  difficult  subject  of  the  Discussion  of  Equations  lias 
been  reserved  till  late  in  the  course,  for  several  reasons.  Thus,  when  the  pupil 
reaches  this  topic,  he  has  become  familiar  with  most  of  the  principles  to  he 
applied,  and  has  Ijecome  sufficiently  imbued  with  the  spirit  of  the  algebraic 
analy.sis  to  be  enabled  to  grasp  it.  To  discuss  an  equation  independently  and 
well,  is  a  high  mathematical  accomplishment,  and  should  not  be  expected  of  tlie 
tyro.  It  is  nothing  else  than  to  think  in  mathematical  formula?,  and  hence  is 
one  of  the  later  products  of  mathematical  study.  It  is  hoped  that  the  position 
assigned  to  this  subject  in  the  course,  and  the  manner  of  treating  it,  will  insures 
better  results  tlian  we  have  hitherto  been  able  to  obtain. 

6.  In  the  selection  of  Subjects  to  be  Presented,  constant  regard  has  been  had 


PREFACE.  V 

to  tlie  demands  of  the  subsequent  nKithcniatical  coarse.  This  has  led  to  the 
omission  of  a  number  of  theorems  and  methods,  which,  though  well  enough 
in  themselves  as  mere  matter  of  theory,  find  no  practical  application  in  a  sub- 
sequent course,  however  extended  ;  and  has,  at  the  same  timy,  led  to  the 
introduction  of  not  a  few  things  which  the  advanced  student  always  finds  occa- 
gion  to  use,  but  for  which  he  searches  his  Algebra  in  vain,  if  he  has  at  hand 
nothing  but  our  common  American  text-books. 

7.  In  Method  of  Treatment  the  following  principles  have  been  kept  constantly 
in  mind  :  1.  That  the  view  presented  be  in  line  with  the  mathematical  thinking 
of  to-day.  2.  That  everything  be  rigidly  demonstrated  and  amply  and  clearly 
illustrated.  3.  When  long  experience  has  shown  that  the  majority  of  good 
students  have  difficulty  in  comprehending  a  subject,  special  pains  should  be 
taken  to  elucidate  it.  4.  No  principle  is  thoroughly  learned  by  a  pupil  until  he 
can  apply  it  ;  and  nothing  so  fixes  principles  in  the  mind  as  the  use  of  them. 
Hence  an  unusually  large  number  of  examples  has  been  introduced.  5.  It  is 
often  necessary  to  multiply  examples  in  order  to  meet  the  requirements  of  the 
class-room. 

8.  Answers. — The  answers  to  examples  are  not  generally  annexed  to  them  in 
the  text.  There  are,  however,  two  editions  of  the  volume,  one  with  the  answers 
at  the  end,  and  the  other  without  any  answers,  except  an  occasional  one  in  the 
body  of  the  book. 

9.  Finally,  the  Order  of  Topics  is  such  that  a  student  requiring  a  less  extended 
course  than  the  entire  volume  presents,  can  stop  at  any  point,  and  feel  assured 
that  what  he  has  studied  is  of  more  elementary  importance  than  what  follows. 
Thus  students  who  do  not  desire  to  study  the  Higher  Equations  can  conclude 
their  course  with  the  first  chapter  of  Part  III.;  and  a  course  which  includes  the 
first  three  chapters  of  this  part  will  be  found  as  extended  as  most  of  our 
Academies,  and  perhaps  many  of  our  Colleges,  will  find  expedient. 

&uch  works  as  those  of  Seiiret,  Cirodde,  Comberousse,  Wood,  Hymers, 
Hind,  Todhunter,  Young,  and  most  of  our  American  treatises,  have  been  at 
hand  during  the  preparation  of  the  entire  volume.  To  WrirrwoRTn's  charming 
little  treatise  on  Choice  and  Chance,  the  author  is  indebted  for  a  number  of 
examples  in  the  last  section. 

The  quick  eye  and  cultivated  taste  of  my  friend,  Mr.  W.  W.  Beman,  AM., 
Instructor  of  Mathematics  in  the  University,  have  done  me  excellent  service  in 
reading  the  proof-sheets,  and  have,  I  trust,  given  the  work  a  dtrgree  of  typo- 
graphical accuracy  not  usually  found  in  first  issues  of  such  treatises. 


Tl  PREFACE. 

Witli  these  words  of  explanation  as  to  wlmt  I  have  attempted  to  do,  I  commit 
tlie  volume  to  the  hands  of  my  f  el  low-laborers  in  the  work  of  teaching,  assured 
from  the  generous  and  ai>preciative  reception  which  they  have  given  my  previous 
efforts,  that  this  will  not  fail  of  a  candid  consideration. 

EDWARD  OLXEY. 

Univeiisity  op  MmiKJAN, 
Ann  Arbot'y  July,  1873. 


^ 


ONTIenTS')) 


INTRODUCTION. 

SECTION   I. 
General  Definitions,  and  the  Algebraic  Notation. 

PAGB 

Branches  of  Pure  Mathematics: 

Pure  Mutheiuatics. — Definitiou  (1) ;  branches  enumerated  (3,  3) 1 

Quantity.— Definition  (4) 1 

Number. — Definition  (5) ;  Discontinuous  and  Continuous  (6,  7,  8) 1,  2 

Definition  of  the  several  branches. — Arithmetic  (9) ;  Algebra  (10) ; 
Calculus  (11);  Geometry  (12) 2 

Lo^hco-Mathematical  Terms  : 

Proi)osition. — Definition  (13) 

Varieties  of  Propositions. — Enumerated  (14);  Axiom  (15);  Theo- 
rem (16);  Lemma  (18);  Corollary  (19) ;  Postulate  (20) ;  Problem  (21).  2,  % 

Definition  of.— Demonstration  (17) ;  Rule  (22) ;  Solution  (23) ;  Scho- 
lium (24) 2,3 


PART  L— LITERAL  ARITHMETIC. 


CTIAPTEU  I. 
FUNDAMENTAL  RULES, 


SECTION   I. 

Notation. 

System  of  Notation.— Definition  (25) 4 

Symbols  of  Quantity.— Arabic  (2G) ;  Literal  (27) ;   advantages  of  latter  (28).  4,  5 

l^awH.— Of  Decimal  Notation  (29) ;  Of  Literal  Notation  (30,  31) 5 

Symlx)l  20  ,  and  its  meaning  (32) «*> 

Symbols  of  Operation  (33) 6 

Definitions.—Exponent  (e34);    A  Positive  Integer  (35);   a    Positive  Frac 

tion  (3G) ;  a  Negative  Number  (38) ;  Radical  Sign  (37). . .   fl 


via  CONTENTS. 

PAOS 

Symbols    of    Relation.— Sign  :  (39);     Sign  ..  (40);     Signs  =,  : :   (41); 

Sign  a  (42) ;  Signs  <  >  (43) 6,  7 

Synilx)la  of  Aggregation  ,  (),[],    |  1 ,    |  ,  (44,  45) 7 

Symbols  of  (Continuation  (46) 7 

Synib<jls  of  Deduction  (47) 7 

Positive  and  Negative. — How  applied  (48) ;  Double  use  of  signs  +  and 
—  (4U) ;  Essential  sign  of  a  quantity  (50) ;  Illustrations  ;  abstract 
quantities  no  sign  (51) ;  less  than  zero  (52) ;  increase  of  a  nega- 
tive quantity  (53) 7-9 

Names  of  Different  Forms  of  Expre.s.sion. — Polynomial  (54) ;   Monomial, 

Binomial,  Trinomial,  etc.  (55) ;  Coefficient  (56) ;  Similar  Terms  (57) .     9. 10 


SECTION   II. 

Addition. 

Definitions.— Addition  (58) ;  Sum  or  Amount  (59) 10 

Prop.  1.— To  add  similar  terms  (60)  ;  Cor.  1.  Sign  of  the  sum  (61)  ;  Sch. 
Addition  sometimes  seems  like  Subtraction  ;  Cor.  3.  Algtibraic 
sum  of  a  i)ositive  and  negative  quantity  (62) ;    Cor.  3.  Addition 

does  not  always  imply  increase  (63) 10,  11 

Prop.  2. — To  add  dissimilar  terms  (64) ;  Sch.  The  sign  —  before  a  term  ; 

Cor.  Adding  a  negative  quantity  (65) 11,  12 

Prob. — To  add  polynomials  (66) 12 

Prop.  3.— Terms  but  partially  similar  (67) 12 

Prop.  4. — Compound  terms  (68) ;  Sch.  Examples 12,  14 


SECTION   III. 

S  U  R  T  11  A  C  T  I  O  N. 

Definfttons.— Subtraction  (69)  ;  Difference  (70) 14 

Prou. — To  perform  Subtraction  (71)  ;  Cor.  1.  Removing  a  i)arcntlicsis 
preceded  by  the  sign  —  (72) ;  Cor.  2.  Introducing  a  paren- 
thesis (73) ;    Examples  ;  Theory  of  Subtraction 14-16 


SECTION    IV. 
MuiiTII'MrATJON. 

Definition.— (74) ;  Corft.  1  and  2.  Character  of  factors  and  product  (75, 76)         16 

Prop.  1.— Order  of  factors  immaterial  (77) 16 

Prop.  2.— Sign  of  Product  (78) ;    Com.  1,  2,  and  3.  Sign  of  product  of 

several  factors  (79-81) 16, 17 

Prop.  3.— Product  of  quantities  afFecte*!  with  exponents  (82) ;  Examples ; 

Sell.  Fractional  exponent 17,  18 

Prob.— To  multiply  monomials  (83) 18,  19 

Prod.— To  multiply  polynomials  (8-1) 19 


CONTENTS.  IX 

PAQE 

Three  Important  Theorems. — Square  of  the  sum  (85)  ;  Square  of  the  dif- 
ference (8(5)  ;  Product  of  sum  and  difference  (87)  ;  Examples. ...   19,  20 
Multiplication  by  Detached  Coetiicients  (88) ;  Examples 21 


SECTION   V. 

Division. 

Definitions. — Division  (89) ;    Problem,  how  stated  (90) ;    Cars.  1-5.  De- 
ductions from  definition  (91-95)  ;  Cancellation  (96) 22 

Lemm.v  1.— Sign  of  Quotient  (97) 22 

Lemma  2.— a"'  -»-  a"  (98) ;  Cor.  1.  Exponent  0  (99) ;  Cor.  2.  How  negative 
exponents  arise  (100)  ;  Cor.  3.  To  transfer  a  factor  from  dividend 

to  divisor  (101) 22,  23 

Puor..  1. — Division  of  monomials  (102) 23 

Piion.  2. — To  divide  a  polynomial  by  a  monomial  (103) ;  Arrangement  of 

terms  (104) 23 

Pnon.  .J.— To  divide  a  polynomial  by  a  polynomial  (105) ;  ScJi.  Similar  to 

' '  Long  Division  ; "  Examples 24,  25 

Division  by  Detached  Coefficients  (100) ;  Examples 25,  26 

Synthetic  Division  (107) ;  Examples 26,  27 


CHAPTER  II. 
FACTORING. 

SECTION   I. 
Fundamental  Propositions. 

Definitions.— Factor  (108) ;    Common    Divisor  (109) ;    Common    Multi- 
ple (110)  ;  Composite  Number  (111)  ;  Prime  (112,  113) 28 

Prop.  1.-  To  resolve  a  monomial  (114) 29 

Prop.  2. — To  remove  a  monomial  factor  from  a  polynomial  (115) 29 

Prop.  3. — To  factor  a  trinomial  scpiare  (1 10) 29 

Prop.  4.— To  factor  the  difference  of  two  squares  (117) 29 

Prop.  5.— Given  one  factor  to  find  the  olher  (118) 20 

Prop.  G. — By  what  the  sum  and  difference  of  like  j)Ovvers  are  divisible(119)  ; 

Cor.  Prop.  0  applied  to  fractional  and  negative  exponents  (120).  .29,  30 

Prop.  7. — To  resolve  a  trinomial  (121) 

Prop.  8. — To  resolve  a  polynomial    by  separating  it   into  parts  (122) ; 

Examples 31-33 


SECTION   II. 
Greatest  or  Highest  Common  Divisor. 

Definition  (123) 33 

Lemma  1. — The  H.  C.  D.  the  product  of  all  the  common  factors  (124) ; 

Exami)le8 33,  34 


X  CONTENTS. 

PAGB 

Lemma  2.— A  divisor  of  a  polynomial  of  the  form  Ax^  -\  Bx'"-^  +  Ca^-^  + 

+  Ex+  F  (136) 34 

Lemma  3. — A  divisor  of  a  number  divides  any  multiple  (127) 35 

Lemma  4. — A  C.  D.  divides  the  sum  and  also  the  difference  (128) 35 

General  Rule  for  H.  C.  D.  demonstrated  and  applied  (129, 130) 35-38 


SECTION    III. 

Lowest  or  Least  Common  Multiple. 

Definition  (131) 39 

Pkoh. — To  find  the  L.  C.  M.  hy  resolving  numbers  into  factors  (132) ; 

Examples 39,  40 

Finding  the  L.  C.  M.  facilitated  by  the  method  of  H.  C.  D 40 


CHAPTER   III. 

FJi  ACTIONS. 

Definitions  and  Fundamental  Principles  : 

Fraction  (133) ;  The  true  conception  of  a  literal  fraction  (134)  ; 
Corff.  1  and  2.  Chanires  in  terms  of  a  fraction  (135,136)  ;  Integral 
Form,  Mixed  Kuiulxr.  Proi>er  Fraction,  Im])roper,  Simple,  Com- 
pound, Complex,  Lowist  Terms,  Lowest  Common  Denomina- 
tor (137-146) 41,  42 

Reduction.— Definition  (147) ;  Kinds  of  (148) 42,  43 

Prop,.  1.— To  lowest  terms  (149)  ;  Seh,  1.  The  use  of  the  H.  C.  D.; 

Sc/t.  2.  The  converse  process 43 

Prop  2. — From  improper  to  mixed  form  (150);  Cor.  Use  of  nega- 
tive exponents  (151) 43 

Prop.  3. — From  integral  or  mixed  to  fractional  form  (152) 43 

Prop.  4. — To  common  denominator  (153) ;  Cor.  To  L.  C.  D.  (154). .  44 

Prop.  5. — Complex  to  simple  (155) 44 

Addition  : 

Prop. — To  add  fractions  (150)  ;  Co7'.  To  add  mixed  nunilwrs  (157).  44,  45 

SURTRACTION  : 

Prop.— To  subtract  fractions  (158)  ;  Cor.  To  subtract  mixed  num- 
bers (159) 45 

Multiplication  : 

Prop.  1. — A  fraction  by  an  integer  (100) 45 

Prop.  2. — To  multiply   by  a    fraction  (161);     Cor.    To  multiply 

mixed  numbers  (162) 45,  46 

Division  : 

Prop.  1.— To  divide  by  an  integer  (163) 46 

Prop.  2. — To  divide  by  a  fraction  (164) ;  Reason  for  inverting  the 
divisor ;  Sell.  To  reverse  the  operation  of  multiplication  ;  Cor. 
Reciprocal,  what  (165)    46,  47 


CONTENTS.  XI 

PAGK 

Signs  of  a  Fraction  : 

Three  things  to  consider  (166) 47 

Essential  character  (167) ;  Examples 47-58 


CHAPTER  IV. 
POWERS  AND  ROOTS. 


SECTION  I. 

Involution. 
General  Definitions  : 

Power,  Degree  (168) ;  Root  (169) ;  Exponent  or  Index  ;  How  to  read 
an  Exponent  (170) ;  Radical  Number,  Rational,  Irrational, 
Surd  (171) ;  Radical  Sign  (172) ;  Imaginary  Quantity  (173) ; 
Real  (174) ;  Similar  Radicals  (175) ;  Rationalize  (176) ;  To  affect 
with  an  exponent  (177)  ;  Involution  (178) ;  Evolution  (17i)) ;  Cal- 
culus of  Radicals  (180) 54,  55 

Involution  : 

Pkob.  1. — To  raise  to  any  power  (181) ;  Cor.  Signs  of  powers  (182).  55,  56 

Prob.  2.— To  affect  with  any  exponent  (183) 5(» 

Pros.  3.— The  Binomial  Formula  (184) ;  Cor.  1.  Wlien  the  series 
terminates  (185)  ;  Cor.  2,  Number  of  terms  (186) ;  Cor.  3.  Equal 
Coefficients  (187) ;  Cor.  4.  Sum  of  exponents  in  any  tenn  (188)  ; 
Coi'.  5.  Statement  of  the  Rule  (189) ;  Cor.  6.  Signs  of  the  terms 
in  the  expansion  of  (a—b)'^  (190) ;  Examples 57-60 


SECTION   II. 
Evolution. 

Prob.  1.— To  extract  root  of  perfect  power  (101) ;  Sch.  Signs  of  root  (192) ; 

Cor.  1.  Roots  of  monomials  (193) ;  Cor.  2.  Root  of  Product  (194) ; 

Cor.  3.  Root  of  a  Quotient  (195) ;  Examples 60,61 

Prob.  2, — To   extnu-t   lootn   whose   indices   are   composed  of   factors  2 

and  3  (196) 62 

Prob.  3.— To  extract  the  mih  root  of  a  number  (197) ;  Examples 62-67 


SECTION  in. 

Calculus  of  Radicals. 
Reductions  : 

Pnow.  1  .—To  remove  a  factor  (198) ;  Cor.  To  giniplify  a  fraction  (190)        67 

Pkob.  2.— When  the  index  is  a  composite  number  (200/ 68 

Prob.  3.— To  any  required  index  (201) ;  Cor.  To  put  the  coefficient 

under  the  radical  sign  (202) 68 


XH  CONTENTS. 

PAQK 

Prob.  4.— To  a  common  index  (303) G8 

Prob.  5. — To  rationalize  a  monomial  denominator  (204) 69 

Prob.  6. — To  rationalize  a  radical  binomial  denominator  (205).. ...  00 

Prop.  1. — To  rationalize  any  binomial  radical  (206) 69 

Prop.  2.— To  rationalize  Va-\-  Vb  +  Vc  (207) ;  Examples 70-72 


SECTION  IV. 

Combinations  of  Radicals. 

Addition  and  Subtraction  : 

Prob.  1.— To  add  or  subtract  (208) 73 

Multiplication  : 

Prop.  1.— Product  of  like  roots  (209) 73 

Prop.  2.— Similar  Radicals  (210) 73 

Prob.  2.— To  multiply  Radicals  (211) 73 

Division  : 

Prop.— Quotient  of  like  roots  (212) 73 

Prob.  3.— To  divide  Radicals  (213) 73 

Involltion  : 

Prob.  4. — To  raise  to  any  power  (214) ;  Cor.  Index  of  power  and 
root  alike  (215) 73 

Evolution  : 

Prob.  5. — To  extract  any  root  of  a  Monomial  Radical  (216) 74 

Prob.  6. — To     extract     the     square     root      of      a  ±  n  Vb,     or 
mVa±  7it^(217);  Examples 74-76 


SECTION   V. 

Imaginary  Quantities. 

Definition  (218) ;  not  unreal  (219) ;  a  curious  property  of  (220) 76 

Prop.— Reduced  to  form  m^^^^X  (221) ;  fkh,  Tlie  form  mV^^l  (222)..  77 
Prob. — To  add  and  subtract  imaginary  monomials  of  second  degree  (223); 

Examples 77,  78 

Prop.— Polynomial  reduced  to  form  a±&^^l(224);    Sch.  Conjugate 

Imaginaries,  Modulus  (225) ;  Examples 79 

Multiplication  and  Involution  : 

Prob.— To  determine  character  of  product  (226) ;  Examples 70,  80 

Division  op  Imaginaries  : 

Prob. — To  divide  one  imaginary  by  another  (227) ;  Examples 80,  81 


CONTENTS.  XI 11 


PART  IL— ELEMENTARY  COURSE  IN  ALGEBRA. 


CHAPTER  I. 
SIMPLE  EQUATIONS. 


SECTION  I. 

Equations  with  one  Unknown  Quantity. 
Definitions  :  pagb 
Equation  (1) ;  Algebra  (3) ;   Members  (3) ;  Numerical  Equation  (4) ; 
Literal  Equation  (5) ;   Degree  of  an  Equation  (6) ;   Simple  Equa- 
tion (7) ;  Quadratic  (8) ;  Cubic  (9)  ;   Higher  Equations  (10) 82,  83 

Transformations  : 

What  (11, 12) ;   Axioms  (13) 83 

Prob, — To  clear  of  Fractions  (14) ;  Transposition  (15) 83 

Prob. — To  transpose  (16) 84 

Solution  of  Simple  Equations  ; 

What  (17) ;  When  an  equation  is  satisfied  (18) ;  Verification  (19). . .  84 
Prob,  1. — To  solve  a  simple  equation  (20) ;  8ch.  1,  Kinds  of  changes 
which  can  be  made  (21);  Car.  1.  Clianging  signs  of  both  mem- 
bers (22) ;  8ch.  2.  Not  always  expedient  to  make  the  transforma- 
tions in  the  same  order  (23)  ;  Sch.  3,  Equations  which  become 
simple  by  reduction  (24) 85 

Simple  Equations  containing  Radicals  : 

Prob.  2.— To  free  an  equation  of  Radicals  (25,  26) 85 

Summary  of  Practical  Suggestions  (27,  28) ;  Examples ,  86-88 

Applications  to  the  Solution  of  Exampi.es  (29)  ;  Statement,  Solu- 
tion (30) ;  Knowledge  required  in  making  statement  (31) ;  Direc- 
tions to  guide  in  making  statement  (32) ;  Not  always  best  to  use 
X  (33)  ;  Examples 89-92 


SECTION  IT, 

Independent,   Simultaneous,  Simple  Equations  with  two  Unknown 

Quantities, 
Definitions  : 

Independent  Equations  (34)  ;  Simultaneous  Equations  (35) ;  Elimi- 
nation (36) ;  Methods  (37) 93 

Elimination  : 

Prob.  1.— By  Comparison  (38) 93 

Prob.  2.— By  Substitution  (39) 94 


XIV  CONTENTS. 

PAGE 

Prob.  3.— By  Addition  or  Subtraction  (40) 9 i 

PROB.  4. — By  Undetermined  Multipliers  (41) 95 

Prob.  6. — By  Division  (43) ;  Examples  and  Applications 96-99 


SECTION   III. 

Independent,  Simultaneous,  Simple  Equations  with  more  than  two 
Unknown  Quantities. 

Prob.— To  solve  (43) ;  Examples  and  Applications 1(X)-103 


CHAPTER  II. 
RATIO,  PROPORTION,  AND    PROGRESSION. 


SECTION  I. 
Ratio. 

Definitions — Ratio  (44) ;  Sign  (45) ;  Cor.  Effect  of  Multiplying  or 
Dividing  the  Terms  (40) ;  Direct  and  Reciprocal  Ratio  (47) ; 
Greater  and  I^ss  Inequality  (48) ;  Compound  Ratio  (49) ;  Du- 
plicate, Subduplicate,  etc.  (50) 104,  105 

Examples 105,  106 


SECTION  II. 

Proportion.  ^ 

Oepinitions.— Proportion  (51) ;  Extremes  and  Means  (52) ;  Mean  Pro- 
portional (53) ;  Third  Proportional  (54) ;  Inversion  (55)  ;  Alter- 
nation (56) ;  Composition  (57) ;  Division  (58) ;  Inversely  Pro- 
portional (59) ;  Continued  Proportion  (60) 106,  107 

Prop.  1. — Product  of  extremes  equals  product  of  means  (61)  ;  Cor.  1. 

Square  of  mean  proportional  (62);  Cor.  2.  Value  of  any  term  (63)  107 

Prop.  2.— To  convert  an  equation  into  a  projMjrtion  (64) ;  Cor.  Taken 

by  alternation  and  inversion  (65) 107,  108 

Prop.  3. — What  transformations  can  be  made  without  destroying  the 

proportion  (66,  67) 108 

Prop.  4. — Products  or  Quotients  of  corresponding  terms  of  two  pro* 

portions  (68) ;  Cor,  Like  powers  or  roots  (69) 108 

Prop.  5. — Two  proportions  with  equal  ratio  in  each  (70) 108 

Prop.  6. — Taken  by  composition  and  division  (71) ;  Cor.  Series  of  equal 
ratios  as  a  continued  proportion  (72) ;  Sch.  Method  of  testing 
any  transformation  (73)  ;  Examples  and  Applications 109-113 


CONTENTS.  XV 

SECTION  III. 
Progressions. 

PAGE 

Definitions. — Progression,  Arithmetical,  Geometrical,  Ascending,  De- 
scending, Common  Difference,  Ratio  (74) ;  Signs  and  Illustra- 
tions (75) ;  Arithmetical  and  Geometric  Mean  (76,  77)  ;  Five 
things  considered  (78) .♦. 113,  114 

Arithmetical  Progression. — Prop.  1.  To  find  the  last  term  (79) ; 
Prop.  2.  To  find  the  sum  (80) ;  Cor.  1.  These  formulas  sutE- 
cient  (81) ;  Cor.  2.  To  insert  means  (82) ;  Formulae  in  Arith- 
metical Progression  (83) ;  Examples 114-117 

Geometrical  Progression. — Prop.  1.  To  find  the  last  term  (84) ; 
Prop.  2.— To  find  the  sum  (85) ;  Cor.  1.  These  formulas  suflS- 
cient  (86) ;  Cor.  2.  Another  formula  for  sum  (87) ;  Cor.  3.  To 
insert  means  (88) ;  Cor.  4.  Sum  of  an  infinite  series  (89) ;  Geo- 
metrical Formulae  (90) ;  Examples 117-122 

SECTION   IV. 

Variation. 

Definitions. — Variation,  directly,  inversely,  jointly,  directly  as  one 
and  inversely  as  another  (91-93) ;  Sign  (94)  ;  Prop.  Variation 
expressed  as  Proportion  (95) ;  Exercises 123-125 


SECTION  V. 

Harmonic  Proportion  and  Progression. 
Definitions. — Harmonic    Proportion   (96) ;       Harmonic    Mean   (97) ; 
Prop.  Quantities  in  Harmonic  Proportion,  their  Reciprocals  in 
Arithmetical  (98) ;   Harmonic  Progression  (99) ;   Derivation  of 
the  term  Harmonic  (100) ;  Exercises 125,  126 


CHAPTER  III. 
QUADRATIC  EQUATIONS. 


SECTION  I. 

Pure  Quadratics. 

Definitions.— Quadratic  (101);  Kinds  (102);  Pure  (103);  Affected  (104); 

Root  (105) 127 

Resolution  of  a  Pure  Quadratic  Equation  (106) ;  Cor.  1.  A  Pure 

Quadratic  has  two  roots  (107) ;  Cor.  2.  Imaginary  roots  (108). .  127,  128 
Examples  and  Applications 128-130 


XVI  CONTENT& 

SECTION   II. 
Affected  Quadratics. 

PAGE 

Definition  (109) 130 

Resolution. — Commou  Method  (110) ;  Sch.  1.  Completing  the  Square  ; 
Cor.  1.  Two  roots,  character  of  (111) ;  Cor.  2.  To  write  the 
roots  of  x'^  +  px  =  q,  witliout  completing  the  square  (112) ; 
Cor.  3.  Special  methods  (113, 114) ;  Examples 130-134 


SECTION   III 

Equations  of  other  Degrees  which  may  be  Solved  as  Quadratics. 

Prop.  1. — Any  pure  equation  (115) 134 

Prop.  2. — Any  equation  containing  one  unknown  quantity  with  only 

two  different  exponents,  one  of  which  is  twice  the  other  (116).  135 

Prop's  3-5.— Special  Solutions  (117-122) ;   Examples 135-139 


SECTION   IV. 

Simultaneous  Equations  of  the  Second  Degree  between  two 
Unknown  Quantities. 

Prop.  1. — One  equation  of  the  second  degree  and  one  of  the  first  (123).  140 

Prop.  2. — Two  equations  of  the  second  degree  usually  involve  one  of 

the  fourth,  after  eliminating  (124) 140 

Prop.  3.— Homogenous  quadratics  (125,  126) 140,  141 

Prop.  4. — When  the  unknown  quantities  are  similarly  involved  (127). .  141 

Examples,  Special  Solutions,  Applications 142-147 


CHAPTER   IV. 

INEQUALITIES. 

Dephtition  (128)  ;  Fundamental  Principle  (129) ;  Members  (130) ;  Same 
transformations     as     equations    (131)  ;     Same     and     opposite 

sense  (132) 148 

Prop. — Sense  of  an  inequality  not  changed  (133) 148,  149 

Prop. — Sense  of  an  inequality  changed  (134) ;  Exercises 149,  150 


CONTENTS.  XVll 


PART  IIL— ADVANCED   COURSE   IN  ALGEBRA. 


CHAPTER  I. 
INFINITESIMAL  ANALYSIS, 


SECTION  I. 
Differentiation. 

PAG« 

Definitions.— Constant  and  Variable  Quantities  (135-137) ;  Sch.  Dis- 
tinction between  constant  and  variable,  known  and  un- 
known (138) ;  Function  (139)  ;  How  represented  (140) ;  Inde- 
pendent and  Dependent  Variable  (141);  Infinitesimal  (142); 
Consecutive  values  (143)  ;  Differential  (144) ;  Notation  (14."j) ; 
To  differentiate  (14G) 151-153 

Rules  for  Differentiating  : 

Rule  1. — To  differentiate  a  single  variable  (147) 154 

Rule  2.— Constant  factors  (148) 154 

Rule  3. — Constant  terms  (149) 151 

Rule  4. — The  sum  of  several  variables  (150) 155 

Rule  5. — The  product  of  two  variables  (151) 155 

Rule  C. — The  product  of  several  variables  (152) 155,  15G 

Rule  7. — Of  a  fraction  with  a  variable  numerator  and  denomina- 
tor (153) ;    Cor.  With  constant  numerator  (151) ;    Sch.  Constant 

denominator  (155) 150 

Rule  8 — Of  a  variable  affected  with  an   exponent  (15G) ;    Sch. 

Rate  of  change  (157) ;  Examples 156-158 


SECTION   II. 

Indeterminate  Coefficients. 

Definition  (158) 159 

Prop.— In  A  ^^  Bx  -\-  Gx^  +  etc.  =  ^'+  B'x  -i-  C'x"^  +  etc.,  coefficients 
of  like  powers  of  x  equal  to  each  other  (159) ;  Cor.  A,  B,  C, 
etc.,  =  0  (160) 150 

Development  of  Functions  (161) ;  Examples 159-10^ 

Decomposition  of  Fractions  (163) ;  Cme  1.  When  the  denominator 
is  resolvable  into  re(jl  and  unequal  factors  of  the  firrit  de 
gree  (164)  ;  Cane  2.  Into  real  and  equal  factors  of  the  first  de- 
gree (165) ;  Ca«e  3.  \\\io  real  and  quadratic  factor^  (lOG)  ;  Sch. 
Forms  combined  (167) ;  Examples 102-164 


XVlll  CONTENTS. 

SECTION   III. 
The  Binomial  Formula. 

PAOB 

Binomial  Theorem  (168) ;  Cor.  1.  The  general  term  (169) ;  Scale  of 
Relation  (170) ;  Cor.  2.  Formula  for  scale  of  relation  (171) ; 
Examples 165-168 


SECTION  IV. 
Logarithms, 

Definitions.— Logarithms,  Base  (172) ;  Cor.  Logarithm  of  1  (173) ;  Sys- 
tem of  Logarithms  (174) ;    Two  in  use  (175) ;    Cor.  Quantities 

that  cannot  be  used  as  a  base  (176) 168,  169 

Prop.  1. — Logarithm  of  product  equals  sum  of  logarithms  (178) 169 

Prop,  2. — Logarithm  of  quotient  equals  difference  of  logarithms  (179).  170 

Prop,  3. — Logarithm  of  a  power  (180) 170 

Prop,  4,— Logarithm  of  a  root  (181) 170 

Loirarithras  of  most  numbers  not  integral  (182) 170 

("liaructeristic  and  Mantissa  (183) 170 

Prop, — Mantissa  of  decimal  fraction  or  mixed  number  (184) ;    Cor.  1. 
Characteristic  of   any   number  (185) ;    Cor.  2.    Logarithm  of 

0(186) 170,171 

Computation  op  Logarithms  : 

Modulus  (187) ;  Prop.  Differential  of  a  logarithm  (188) 172 

Prob. — To  produce  the  logarithmic  series  (189) ;  Cor.\.  Loga- 
rithms of  same  number  in  two  different  systems,  as  moduli  (190); 
Cor.  2,  To  find  logarithm  of  a  number  in  any  system  know- 
ing the  modulus,  and  also  to  find  modulus  (191) 173,  174 

Prob. — To  obtain  series  for  computing  Napierian  logarithms  (192)  175 
Prob. — To   compute   Napierian    logarithms    of    natural    num- 
bers (193) 175 

Prop, — Modulus  of  common  system  (194) 1 76 

Tables  of  Logarithms.— What  (195) 177 

Prob,— To  find  the  logarithm  of  a  number  (196,  197) 177,  178 

Prob. — To  find  a  number  corresponding  to  a  logarithm  (198)  . . .  178 

Prop.— The  Napierian  base  (199) ;  Examples 179-181 


SECTION   V. 
Successive  Differentiation  and  Differential  Coefficients. 

Prop. — Differentials  not  necessarily  equal  (200)  ;  Cor.  dy  a  varia- 
ble (201) ;  Notation  (202) ;  dx  constant  (203)  ;  Second  and  Third 
Differentials  (204) ;  Examples  181-183 

Differential  Coefficients  : 

First  Differential  Coefficient  (205) ;  Second  Differential  Coeffi- 
cient (206) ;  Examples ;  Successive  coefficients  written  by  in- 
spection (207) 183-185 


CONTENTS.  XIX 


SECTIC:^  VI. 
Taylor's  Formula. 

PAGB 

Definition  (208) :    Partial  Differential  Coefficients  (209) :    Lemma. 


^    and   ~    equal(210)    185,186 

Prob. — To  produce  Taylor's  Formula  (211) ;  Sch.  First,  second,  etc., 
terms  (212) ;  To  develop  a  function  of  a  variable  witli  an  in- 
crement (213) ;  Examples 186-189 


SECTION  VII. 

Indeterminate  Equations. 

Definition,  Nature,  etc.  (214-220) ;  Examples 189-195 


CHAPTER  11. 

LOCI  OF  EQUATIONS. 

Pkop.  — Every  equation  between  two  variables  may  represent  a  line  (221)  196 

Definitions. — Axes   of    Reference,  Abscissa,  Ordinate,  Co-ordinates 

(222) ;  Locus,  Constructing  Locus  (223) ;  Examples 198-202 

Prob. — To  construct  real  roots  of  equations  witli  one  unknown  quan- 
tity (224) ;  Examples 202,  203 


CHAPTER  HI. 
HIGHER  EQUATIONS. 


SECTION  I. 

Solution  op  Numerical  Higher  Equations  having  Commensurable 
OR  Rational  Roots. 

No  general  method  of  solution  (226) ;  Real,  commensurable  roots  found 

with  little  difficulty  (227) 203 

Prop. — Transforming  an  equation  into  the  form  .t«  -t-  ^4.?"-'  h-  Bx!^-^ 

+  Cx^'3 X  =  0  (228) ;   Examples 204,  205 

Prop. — Roots  of  an  equation  factors  of  absolute  term  (230) ;  If  <^  is  a 

root,  f{x)  divisible  by  {x  —  a),  and  converse  (231) 206 

Prop.— Wliat  equation  can  have  no  fractional  root  (232,  233) 206,  207 


XX  CONTENTS. 

TAan 
Prop. — Equation  of  nth  degree  has  n  roots  (334) ;  Cor.  1.  /(i)  =  (x—a) 

{x  —  b){x  —  c)  -  •  -  •  {x  —  n),  when  a,  b,  c  -  -  -  -  n  are  roots  of 
f{x)  -  0  (SJJo) ;  Cor.  2.  f{x)  can  have  equal  roots  (236) ;  Cor.  3. 
Imaginary  roots  enter  in  pairs  (237) ;  Cor.  4.  Number  of  real 
roots  in  e(iuations  of  odd  and  even  degrees  (238) ;  Limits  of 
imaginary  roots  (239) ;  Sch.  1.  Proposition  illustrated  geome- 
trically (240) ;  Sc/i.  2.  Imaginary  roots  entering  in  pairs  illus- 
trated (241) 207-209 

Prop. — Method  of  finding  equal  roots  (242) ;    Sc7i.  Sometimes  conve- 
nient to  apply  process  several  times  (243) 209,  210 

Prop.— Cliange  of  sign  in  f{x)  (244) ;  How  illustrated  by  loci  (245) 211 

Prop.— Clianging  signs  of  roots  of  f{x)  (246) ;  Coi:  Another  method  (247)  212 

Prob.— To  evaluate  f(x)  for  x  =  a  (248) 212,  213 

Prob. — To  find  commensurable  roots  of    numerical  higher  equations 

(249);  Examples 213-216 

To  produce  an  equation  from  its  roots  (250) ;   Examples 216 


SECTION   II. 

Solution  of  Numerical  Higher  Equations  having  Real,  Incommen- 
surable, OR  Irrational  Roots. 

Typical  form  of  equation  (251) ;  Best  general  method  (252) 216,  217 

Sturm's  Theorem  and  Method  : 

Definition  and  Object  (253,  254) ;  Sturmian  Functions  (255) ;  No- 
tation (256) ;  Permanence  and  Variation  (257) 217,  218 

Sturm's  Theorem  (258) ;  Cor.  1.  To  find  the  number  of  real  roots 
of  f{x)  (259) ;  Cor.  2.  To  find  the  number  of  real  roots  of  f{x) 
between  a  and  b  (260) ;  Sch.  Number  of  imaginary  roots  known 
by  implication  (261) 219-221 

Prob. — To  compute  the  numerical  values  of  f{x),  f\x),  fi{x),  etc. 
(262) ;  Sch.  2.  Usually  unnecessary  to  find  /n(t")  (263) ;  Sch.  3. 
Wlien  the  equation  has  equal  roots  (264) ;  Sell.  4.  Generally 
the  change  of  sign  in  f{x)  enables  us  to  determine  situation  of 
roots  more  easily  than  Sturm's  Theorem  (265) ;  Sch.  5.  Not  ne- 
cessary that  the  coeflficients  should  be  integral  (266) ;  Exam- 
ples     221-228 

Horner's  Method  of  Solution  :  Object  (267) 228 

Prob. — To  transform  an  equation  into  another  with  roots  less  by 

a  (268) ;  Sch.  Signification  of  result  (269) 229 

Prob. — To  compute  the  numerical  values  of  f{a),  /'(a),  hf"{a), 
etc.  (270) ;  Examples 229-233 


CONTENTS.  XXI 

rXOB 

Prop. — Value  of  a;,,  when  a  +  x^  is  a  root  of  /(«)  =  0  (271) 23^235 

Horner's  Rule  (272) ;  jScJioliums  (273-278)  ;  Examples 235-247 


SECTION  III. 

General  Solution  of  Cubic  and  Biquadratic  Equations. 

Cardan's  Solution  of  Cubic  Equations  : 

Prob.— To  resolve  a;^  +  px^  +  ga;  +  r  -  0  (279) 248,  249 

Prop. — Solution    satisfactory    and   unsatisfactory,    when  (280) ; 

Scholium.  Apparently  9  roots  (281) ;  Examples 249-251 

Descartes's  Solution  of  Biquadratics  : 

Prob. -To  resolve  x^  +  ax""  +  bx^  +  dx  ■\- e  =  0  (282) ;  8ch.  In- 
volves solution  of  acubic(283) ...  251,  252 

Recurring  Equations  : 

Definition     (284) 253 

Prop.  1. — The  roots  reciprocals  of  each  other  (285) ;  Sch.  Recip- 
rocal Equations  (286) ;  Cor.  1.  Corresponding  coefficients  with 
like  or  unlike  signs  (287)  ;    Cor.  2.  Reduced  to  form  having 

first  coefficient  unity  (288) 252,  253 

Prop.  2.    Of  an  odd  degree  have  roots  —  1,  and  +  1,  when  (289).  253,  254 

Prop.  3. — Of  an  oven  degree  have  same  roots,  when  (290) 254 

Prop.  4. — Of  an  even  degree  above  second  reduced  to  one  of  half 

that  degree  (291) ;  Examples 254,  255 

Binomial  Equations  and  the  Roots  of  Unity  : 

Definition  (292) ;  Examples  and  Scholium  (293) 255,  25G 

lilXPONENTIAL  EQUATIONS  : 

Definition  (294) 256 

Prob.  1.— To  solve  a""  =  m  (295) 256 

Prob.  2.— To  solve  x^  =  m  (296) ;  Examples 256-260 


CHAPTEll  IV. 
DISCUSSION,  OR  INTERPRETATION,  OF  EQUATIONS. 

Definition  (297) 260 

Prop.— Statement  of  Principles  (298) 260-262 

Real  Number  or  Quantity  (299) 262 

Imaginary  Number  (300) ;   Examples 263-266 

Arithmetical  Interpretation  of  Negative  and  Imaginary  Re- 
sults (301) ;   Sch.  Symbol  ^  (302) ;  Examples 267-271 


XXU  CONTENTS. 


APPENDIX, 


SECTION  I. 
Series. 


PAOB 


Definitions.— Series,  Tenn  (303) ;  Recurring  Series,  Scale  of  Rela- 
tion (304) ;  Infinite,  Convergent,  Divergent  (305) ;  To  revert  a 
Series  (306) ;  Orders  of  Differences  (307) ;  Interpolation  (308) ; 
Enumeration  of  Problems  (309) 273-374 

Lemma. — First  term  of  any  order  of  differences  (310) ;  Cor.  Number  of 

terms  necessary  (311) ;  Examples 274,  275 

Prob.  1. — To  find  scale  of  relation  in  recurring  scries  (312) ;  Sch.  De- 
pendence on  too  many  or  too  few  terms  (313) ;   Examples 275-277 

Prob.  2.— To  find  the  nth  term  (314) ;  Examples 277,  278 

Prob.  3. — To  determine  whether  a  series   is  convergent  or  divergent 

(315) ;  Examples 278-280 

Prob.  4.— To  find  sum  of  a  series  (316) ;  Examples 280-285 

Piling  Balls  and  Shells  (317) 285 

Prop. — Number    in    triangular  pile  (318);      Cor.   Number  of 

courses  (319) 285,  286 

Prop. — Number  in  square  pile  (330);  Cor.  Number  of  courses  (321)  286 

Prop. — Number  in  oblong  pile  (332) ;  Examples 286,  287 

Reversion  of  Series  : 

Prob.— To  revert  a  series  (323) ;   Examples 287,  288 

Interpolation  : 

To  interpolate  between  functions  (324) ;  Sch.  1.  Result  correct 
when  (325) ;  Sell.  2.  Another  formula  (326)  ;  Sch.  3.  Used  in 
Astronomy  (327) ;  Examples 288-291 


SECTION  II. 

Permutations. 

Depinitions.— Combinations  (328);  Permutations  (329);  Arrange- 
ments (330) 393 

Prop.— Number  of  arrangements  of  m  things  n  and  n  (331) ;  Cor.  1. 
Permutations  of  m  things  (333) ;  Cor.  8.  When  p  things  are 
alike,  etc.  (333);  Cor.  3.  Combinations  of  m  things  n  and 
n  (334) ;  Examples 292-294 

Probabilities  :  . 

Mathematical  Probability  and  Improbability  (335) ;  Examples..    294-398 


(3 


SECTION  L 
GENERAL  DEFINITIONS,  AND  THE  ALGEBRAIC  NOTATION. 


BRANCHES  OF  PURE  MATHEMATICS. 

1.  Pure  Mathematics  is  a  generjil  term  applied  to  several 
branches  of  science,  which  have  for  their  object  the  investigation  of 
the  properties  and  relations  of  quantity — comprehending  number, 
and  magnitude  as  the  result  of  extension — and  of  form. 

2.  The  Several  Branches  of  Pure  Mathematics  are  Arith- 
metic, Algebra,  Calculus,  and  Geometry. 

3.  Arithmetic,  Algebra,  and  Calculus  treat  of  number,  and  Geo- 
metry treats  of  magnitude  as  the  result  of  extension. 

4.  Quantity  is  the  amount  or  extent  of  that  which  may  be 
measured;  it  comprehends  number  and  magnitude. 

The  term  quantity  is  also  conventionally  applied  to  symbols  used 
to  represent  quantity.  Thus  25,  m,  xi,  etc.,  are  called  quantities, 
although,  strictly  speaking,  they  are  only  representatives  of  quantities. 

5.  Number  is  quantity  conceived  as  made  up  of  parts,  and 
answers  to  the  question,  "  How  many  ?  " 

6.  Number  is  of  two  kinds.  Discontinuous  and  Continu- 
ous, 

7.  Disco7itinuous  Number  is  number  conceived  as  made 
up  of  finite  parts;  or  it  is  number  which  passes  from  one  state  of 
value  to  another  by  the  successive  additions  or  subtractions  of  finite 
units ;  i.  e.,  units  of  appreciable  magnitude. 

8^  Continuous  Number  is  number  which  is  conceived  as 
composed  of  infinitesimal  parts;  or  it  is  number  which  passes  from 

1 


j2«;|  f^]  t'  INTRODUCTION. 

one  state  of  value  to  another  by  passing  through  all  intermediate 
values,  or  states. 

0,  Arithmetic  treats  of  Discontinuotis  Number,— oi 

its  nature  and  properties,  of  the  various  methods  of  combining  and 
resolving  it,  and  of  its  application  to  practical  affairs. 

10.  Alf/ebra  treats  of  the  Equatioiiy  and  is  chiefly  occupied  in 
explaining  its  nature  and  the  methods  of  transforming  and  reducing 
it,  and  in  exhibiting  the  manner  of  using  it  as  an  instrument  for 
matliematical  investigation.* 

H.  Calculus  treats  of  Contijiuous  Niimher,  and  is  chiefly 
occupied  in  deducing  the  relations  of  the  infinitesimal  elements  of 
such  number  from  given  relations  between  finite  values,  and  the  con- 
verse process,  and  also  in  pointing  out  the  nature  of  such  infinites- 
imals and  the  method  of  using  them  in  mathematical  investigation. 

12,  Geometry  treats  of  magnitude  and  form  as  the  result  of 
extension  and  position. 


LOGICO-MATHEMATICAL    TERMS. 

13.  A  I^roposition  is  a  statement  of  something  to  be  con- 
sidered or  done. 

14.  Propositions  are  distinguished  2kS  Axioms,  Theorems,  Lemmas, 
Corollaries,  Postulates,  and  Problems. 

15.  An  Axiom  is  a  proposition  which  states  a  principle  that 
is  so  simple,  elementary,  and  evident  as  to  require  no  proof. 

10.  A  Tlieorem  is  a  proposition  which  states  a  real  or  supposed 
fact,  whose  truth  or  falsity  we  are  to  determine  by  reasoning. 

17*  A  Danonstration  is  the  course  of  reasoning  by  means 
of  which  the  truth  or  falsity  of  a  theorem  is  made  to  appear.  The 
term  is  also  applied  to  a  logical  statement  of  the  reasons  for  the 
processes  of  a  rule.  A  solution  tells  hoiu  a  thing  is  done  ;  a  demon- 
stration tells  why  it  is  so  done.   A  demonstration  is  often  called  proof. 

*  The  common  definition  of  Algebra,  which  makes  its  distin^nishing  feature;*  to  be  the  literal 
notation,  and  the  me  of  ttie  signs,  is  entirely  at  fault.  When  Algebra  firct  appeared  in  Europe,  it 
possessed  neither  of  these  features !  What  was  it  then  ?  On  the  other  hand,  the  signs  are 
common  to  all  branches  of  mathematics,  and  the  literal  notation  is  as  prominent  in  the  Calculus 
sm  in  Algebra,  and  is  used,  more  or  les?,  in  common  Arithmetic  and  Geometry. 


LOGICO-MATHEMATICAL  TERMS.  3 

1S»  A.  Lemma  is  a  theorem  demonstrated  for  the  purpose  of 
using  it  in  the  demonstration  of  another  theorem. 

19,  A  Corollary  is  a  subordinate  theorem  which  is  sug- 
gested, or  the  truth  of  which  is  made  evident,  in  the  course  of  the 
demonstration  of  a  more  general  theorem,  or  which  is  a  direct 
inference  from  a  proposition. 

20,  A  Postulate  is  a  proposition  which  states  that  something 
can  be  done,  and  which  is  so  evidently  true  a^  to  require  no  process 
of  reasoning  to  show  that  it  is  possible  to  be  done.  We  may  or  may 
not  know  how  to  perform  the  operation. 

21,  A  Prohletn  is  a  proposition  to  do  some  specified  thing, 
and  is  stated  with, reference  to  developing  the  method  of  doing  it. 

22,  A  Rule  is  a  formal  statement  of  the  method  of  solving  a 
general  problem,  and  is  designed  for  practical  application  in  solving 
special  examples  of  the  same  class.  Of  course  a  rule  requires  a 
demonstration. 

23,  A  Solution  is  the  process  of  performing  a  problem  or  an 
example.  It  should  usually  be  accompanied  by  a  demonstration  of 
the  process. 

24,  A  Scholiuin  is  a  remark  made  at  the  close  of  a  discussion, 
and  designed  to  call  attention  to  some  particular  feature  or  features 
of  it. 


PART  L* 

LITERAL    ARITHMETICt 


CHAPTER  I. 

FUNDAMENTAL,    RULES. 


SECTION  L 


NOTATION. 

25.  A  System  of  Notation  is  a  system  of  symbols  by  means 
of  whicli  quantities,  the  relations  between  tliem,  and  the  operations 
to  be  performed  upon  them,  can  be  more  concisely  expressed  than 
by  the  use  of  words. 

Symbols  of  Quantity. 

26,  In  Arithmetic,  as  usually  studied,  numbers  are  represented 
by  the  characters,  1,  2,  3,  4,  5,  6,  7,  8,  9,  0,  called  Arabic  figures,  or, 
simply,  figures. 

27*  In  other  departments  of  mathematics  than  Arithmetic,  num- 
bers or  quantities  are  more  frequently  represented  by  the  common 
letters  of  the  alphabet,  rt,  &,  c,  .  .  .  in,  n,  ,  .  .  x,  y,  z.  These  letters 
may,  however,  be  used  in  Arithmetic ;  and  the  Arabic  figures  ai-e 
used  in  all  departments  of  mathematics.     This  method  of  represent- 

*  Parts  I.  and  IT.  are  a  compend  of  the  elements  of  the  science,  designed  as  a  review  for 
pupils  who  have  studied  some  elementary  treatise,  or  for  the  use  of  such  teachers  and  classes  as 
desire  a  text-book  which  contains  a  condensed  treatment  of  the  subject,  to  be  filled  ont  by  them- 
selves. In  the  author's  Complete  School  Algebra,  the  topics  here  presented  will  be  found 
fully  amplified,  illustrated,  and  applied.  All  the  elementary  principles  are  here  stated,  and  are 
usually  demonstrated.  There  are  also  numerous  examples  under  every  topic.  The  Key  to  the 
Complete  School  Algebra  will  furnish  !idditi<inal  examples  for  use  in  connection  with  this  part, 
t  Part  I.  treats  of  the  familiar  operations  of  Addition,  Subtraction,  Multiplication,  Division, 
Involution  and  Evolution,  and  the  theory  of  Fractions.  The  only  difference  between  the  pro- 
cesses here  developed  and  the  correspond  in"  ones  in  <xymmon  Arithmetic  grows  out  of  the 
notation. 


NOTATION.  5 

ing  quantities  by  letters  is  often  called  the  Algebraic  method,  and 
the  method  by  the  Arabic  characters  the  Arithmetical,  It  would  be 
better  to  call  the  former  the  Literal  method,  and  the  latter  the 
Decimal. 

28,  Tlie  Literal  Notation  has  some  very  great  advantages 
over  the  decimal  for  purposes  of  mathematical  reasoning.  1st,  The 
symbols  are  more  general  in  their  signification ;  and  2d,  We  are 
enabled  to  detect  the  same  quantity  anywhere  in  the  process,  and 
even  iu  the  result.  Thus  it  happens  that  the  processes  become 
general /(9rm?<7cp,  or  rules,  instead  of  special  solutions. 

29.  In  using  the  decimal  notation  certain  laius  are  established,  in 
accordance  with  which  all  numbers  can  be  represented  by  the  ten 
figures.  Thus,  it  is  agreed  tliat  when  several  figures  stand  together 
without  any  other  mark,  as  435,  the  right-hand  figure  shall  signify 
units,  the  second  to  the  left,  tens,  the  third,  hundreds,  etc. ;  also  that 
the  sum  of  the  several  values  shall  be  taken.  This  number  is,  there- 
fore, 4  hundreds  +  3  tens  +  5  (units). 

In  like  manner,  certain  laws  are  observed  in  representing  numbers 
by  letters. 

First  Law. 

30,  Known  Quantities^  that  is  such  as  are  given  in  a  prob- 
lem, are  represented  by  letters  taken  from  the  first  part  of  the 
alphabet;  while  Unknown  Quantities,  or  quantities  whose 
values  are  to  be  found,  are  represented  by  letters  taken  from  the 
latter  part  of  the  alphabet. 

Accented  letters,  as  a',  a",  a'",  a"",  etc.,  (read  "  a  prime,"  "  a  sec- 
ond," "a  third,"  etc.,)  and  letters  with  subscripts,  as  a^,  «,,  a^,  a^^ 
etc.,  (read  ^' a  sub  1,"  "«  sub  2,"  etc.,)  are  sometimes  used.  This 
form  of  notation  is  used  when  there  are  several  like  quantities  in  the 
same  problem,  but  which  have  different  numerical  values.  Thus,  in 
a  problem  in  which  several  walls  of  different  heights,  breadths,  and 
lengths  are  considered,  we  may  represent  the  several  heights  by  a\ 
a",  a",  etc.,  or  a,,  ct.^,  a^,  etc. ;  the  thicknesses  by  b',  b",  b'",  etc.,  or  b^, 
i,,  .^3,  etc.,  and  the  lengths  by  I',  I",  I'",  etc.,  or  Z^,  Z,,  4>  etc. 

The  Greek  letters  are  also  often  used  both  for  known  and  unknown 
quantities. 

Second  Law. 

31.  When  letters  are  written  in  connection,  without  any  sign 
between  them,  their  product  is  signified.  Thus  abc  signifies  that  the 
three  numbers  represented  by  a,  h,  and  c  are  to  be  multiplied  together. 


6  LITERAL  ARITHMETIC. 

32,  A  character  like  a  figure  8  placed  horizontally,  oo ,  is  used  to 
represent  what  is  called  Infinity,  or  a  quantity  larger  than  any 
assignable  quantity. 

Symbols  of  Operation. 

S3.  The  Symbols  of  Operation  used  in  Algebra  are  the 
same  as  those  used  in  Arithmetic,  or  in  any  other  branch  of  mathe- 
matics, and  need  not  be  recapitulated  here. 

EXPONEIJ^TS. 

34»  An  Exponent  is  a  small  figure,  letter,  or  other  symbol 
of  number,  written  at  the  right  and  a  little  above  another  figure, 
letter,  or  symbol  of  number.* 

35.  A  Positive  Integral  Exponent  signifies  that  the 
number  affected  by  it  is  to  be  taken  as  a  factor  as  many  times  as 
there  are  units  in  the  exponent.  It  is  a  kind  of  symbol  of  multipli- 
cation. 

36,  A  JPositive  Fractional  Exponent  indicates  a  power 
of  a  root,  or  a  root  of  a  power.  The  denominator  specifies  the  root, 
and  the  numerator  the  power  of  the  number  to  which  the  exponent 
is  attached. 

57.  The  Radical  Sif/n^  >/,  is  also  used  to  indicate  the 
square  root  of  a  quantity.  When  any  other  than  the  square  root  is 
to  be  designated  by  this,  a  small  figure  specifying  the  root  is  placed 
in  the  sign. 

38,  A  Negative  Exponent,  i.  e.,  one  with  the  —  sign  before 
it,  either  integral  or  fractional,  signifies  the  reciprocal  of  what  the 
expression  would  be  if  the  exponent  were  positive,  ?.  e.,  had  the 
+  sign,  or  no  sign  at  all  before  it. 

Symbols  of  Kelatioi^. 

39,  The  Sign  of  Geometrical  Matio  is  two  dots  in  the 
form  of  a  colon,  :  . 

40,  The  Sign  of  Arithmetical  Matio  is  two  dots  placed 
horizontally,  ••  . 

41,  TJie  Sign  of  Equality  is  two  parallel  horizontal  lines, 
=  .     Tlie  double  colon,  :  : ,  is  the  sign  of  equality  between  ratios. 

*  In  giving  this  definition,  be  careful  and  r<r>/'  add,  "and  indicates  the  power  to  whicli  tho 
number  is  to  be  raised."    This  is  false  :  an  exponent  docs  not  necessarily  indicate  a  power. 


NOTATION.  7 

42,  The  Sign  of  Variation  is  somewhat  like  a  figure  8 
open  at  one  end  and  placed  horizontally,  a  . 

43,  The  Sign  of  Tnequality  is  a  character  somewhat  like 
a  capital  V  placed  on  its  side,  <  ,  the  opening  being  towards  the 
greater  quantity. 

Symbols  of  Aggregation?^. 

44,  A  Vinculum  is  a  horizontal  line  placed  over  several 
terms,  and  indicates  that  they  are  to  be  taken  together.  The  paren- 
thesis, (  ),  the  brackets,  [  ],  and  the  brace,  -j  i  ,  have  the  same 
signification. 

4:5,  A  vertical  line  after  a  column  of  quantities,  each  having  its 

own  sign,  signifies  that  the  aggregate  of  the  column  is  to  be  taken 

as  one  quantity.     Thus,  +  a 

-b 

+  c 


X  is  the  same  as  (ft  —  ^  +  c)x. 


Symbols  of  Continuation. 

46,  A  series  of  dots, ,  or  of  short  dashes, , 

written  after  a  series  of  expressions,  signifies  "  etc."      Thus,  a  :  ar 

:  nr"^  :  ar^ nr"  means  that  the  series   is  to  be  extended 

from  ar^  to  ar",  whatever  may  be  the  value  of  w. 

Symbols  of  Peduction. 

47,  Three  dots,  two  being  placed  horizontally  and  the  thiru 
above  and  between,  .*. ,  signify  therefore,  or  some  analogous  expres- 
sion. If  the  third  dot  is  below  the  first  two,  •.* ,  the  symbol  is  read 
"since,"  "because,"  or  by  some  equivalent  expression. 

Positive  and  Negative  Qiiantities. 

48,  I^ositit^e  and  Negative  are  terms  primarily  applied  to 
concrete  quantities  which  are,  by  the  conditions  of  a  problem, 
opposed  in  character. 

Ill, — A  man's  property  may  be  called  positive,  and  his  debts  negative.  Dis- 
tance up  may  be  called  positive,  aiid  distance  dotcn,  negative.  Time  before 
a  given  period  may  be  called  positive,  and  after,  negative.  Degrees  above  0  on 
the  thermometer  scale  are  called  positive,  and  below,  negative. 

40,  The  signs  -j-  and  —  are  used  to  indicate  the  character  of 
quantities  as  positive  or  negative,  as  Avell  as  for  the  purpose  of  indi- 
cating addition  and  subtraction. 


8  LITERAL   ARITHMETIC. 

50,  In  problems  in  which  the  distinction  of  positive  and  negative 
is  made,  each  quantity  in  the  fonnulw  is  to  be  considered  as  having 
a  sign  of  charade?'  expressed  or  understood  besides  the  plus  or 
minus  sign,  which  latter  indicates  that  it  is  to  be  added  or  sub- 
tracted. The  positive  sign  need  not  be  written  to  indicate  character, 
as  it  is  customary  to  consider  quantities  whose  character  is  not 
specified  as  positive. 

III.  1. — In  the  expression  a'')  +  m  —  e.r,  let  the  problem  out  of  which  it  arose 
be  such,  that  a,  m,  and  x  tend  to  a  positive  result,  and  6  and  c  to  an  opposite,  or 
a  negative  result.  Giving  these  quantities  their  signs  of  character,  we  have 
(  +  fl^)  X  (— &)  +  (  +  wi)  —  (— c)  X  (  +  0-),  which  may  be  read,  "positive  a  mult'- 
plied  by  negative  b,  plus  positive  m,  minus  negative  c  multiplied  by  positive  x." 
Suppressing  the  positive  sign,  this  may  be  written,  a{-h)  +  m  —  {  —  c)x,  by  also 
omitting  the  unnecessary  sign  of  multiplication. 

III.  2. — As  this  subject  is  one  of  fundamental  importance,  let  careful  atten- 
tion be  given  to  some  further  illustrations.  We  are  to  distinguish  between  dis- 
cussions of  the  relations  between  mere  abstract  quantities,  and  problems  in  which 
the  quantities  have  some  concrete  signification.  Thus,  if  it  is  desired  to  ascer- 
tain the  sum  or  difference  of  4C8,  or  m,  and  327,  or  n,  as  mere  numbers,  the 
question  is  one  concerning  the  relation  of  abstract  numbers,  or  quantities.  No 
other  idea  is  attached  to  the  expressions  than  that  each  represents  a  certain  num- 
ber of  units.  But,  if  we  ask  how  far  a  n:an  is  from  his  starting  point,  who  has 
gone,  first,  468,  or  m  miles  directly  east,  a  ul  then  327,  or  n  miles  directly  west ; 
or  if  we  ask  what  is  the  difference  in  time  between  468,  or  m  years  B.  C,  and 
327,  or  n  years  A.  D.,  the  numbers  468,  or  m,  and  327,  or  n,  take  on,  besides  their 
primary  signification  as  quantities,  the  additional  thought  of  opposition  in  direc- 
tion.    They  therefore  become,  in  this  sense,  concrete. 

Again,  a  company  of  5  l3oys  are  trying  to  move  a  wagon.  Three  of  the  boys 
can  pull  75,  85,  and  100  pounds  each  ;  and  they  exert  their  strength  to  move  the 
wagon  east.  The  other  two  boys  can  pull  90  and  110  pounds  each  ;  and  they 
exert  their  strength  to  move  the  wagon  west.  It  is  evident  that  the  75,  85,  and 
100  are  quantities  of  an  opposite  character,  in  their  relation  to  the  problem, 
from  90  and  110.  Again,  suppose  a  party  rowing  a  boat  up  a  river.  Their 
united  strength  would  propel  the  boat  8  miles  per  hour  if  there  were  no  cur- 
rent ;  but  the  force  of  the  current  is  sufficient  to  carry  the  boat  2  miles  per  hour. 
The  8  and  2  are  quantities  of  opposite  character  in  their  relation  to  the  problem. 
Once  more,  in  examining  into  a  man's  business,  it  is  found  that  he  has  a  farm 
worth  m  dollars,  personal  property  worth  n  dollars,  and  accounts  due  him  worth 
c  dollars.  There  is  a  mortgage  on  his  farm  of  h  dollars,  and  he  owes  on  account 
a  dollars.  The  m,  n,  and  c  are  quantities  opposite  in  their  nature  to  h  and  a. 
This  apposition  in  cliaracter  is  indicated  hy  calling  those  quantities  which  con- 
tribute to  one  result  positive,  and  those  which  contribute  to  the  opposite  result 
negative. 

51,  Purely  abstract  quantities  have,  properly,  no  distinction  as 
positive  and  negative;    but,   since  in  such  problems  the  plus  or 


NAMES   OF  DIFFERENT  FORMS  OF  EXPRESSION.  9 

additive,  and  the  minus  or  subtractive  terms  stand  in  the  same 
relation  to  each  other  as  positive  and  negative  quantities,  it  is  cus- 
tomary to  call  them  such. 

III. — In  the  expression  bac  —  Zed  +  %xy  —  2ad,  though  the  quantities,  a,  c,  d, 
X  and  y  be  merely  abstract,  and  have  no  proper  signs  of  character  of  their  own, 
the  terms  do  stand  in  the  same  relation  to  each  other  and  to  the  result,  as  do 
positive  and  negative  quantities.  Thus,  5ac  and  %xy  tend,  as  we  may  say,  to 
increase  the  result,  while  —  'dcd,  and  —  'iad  tend  to  diminish  it.  Therefore  the 
former  may  be  called  positive  terms,  and  the  latter  negative. 

52.  ScH. — Less  than  zero.  Negative  quantities  are  frequently  spoken  of 
as  ''less  than  zero."  Though  this  language  is  not  philosophically  correct, 
it  is  in  such  common  use,  and  the  thing  signified  is  so  sharply  defined  and  easily 
comprehended,  that  its  use  may  possibly  be  allowed  as  a  conventionalism. 
To  illustrate  its  meaning,  suppose,  in  speaking  of  a  man's  pecuniary  affairs, 
it  is  said  that  lie  is  worth  "less  than  nothing;  "  it  is  simply  meant  that  his 
debts  exceed  his  assets.  If  this  excess  were  $1000,  it  might  be  called  nega- 
tive $1000,  or  —$1000.  So,  again,  if  a  man  were  attempting  to  row  a  boat 
up  a  stream,  but  witli  all  his  effort  the  current  bore  him  down,  his  progress 
might  be  said  to  be  less  than  nothing,  or  negative.  In  short,  in  any  case 
where  quantities  are  reckoned  both  ways  from  zero,  if  we  call  those 
reckoned  one  way  greater  than  zero,  or  positive,  we  may  call  those  reckoned 
the  other  way  "less  than  zero,"  or  negative. 

53.  The  value  of  a  Negative  Quantity  is  conceived  to  increase  as 
its  numerical  value  decreases. 

III. — Thus  —3  >  —5,  as  a  man  who  is  in  debt  $3  is  better  off  than  one  who  is 
in  debt  $5,  other  things  l)eing  equal.  If  a  man  is  striving  to  row  up  stream, 
and  at  first  is  borne  down  5  miles  an  hour,  but  by  practice  comes  to  row  so  well 
as  only  to  be  borne  down  3  miles  an  hour,  he  is  evidently  gaining  ;  i.  e.,  —3  is  an 
increase  upon  —5.  Finally,  consider  the  thermometer  scale.  If  the  mercury 
stands  at  20°  below  0  (marked  —20°)  at  one  hour,  and  at  —10°  the  next  hour,  the 
temperature  is  increasing  ;  and,  if  it  increase  suHiciently,  will  become  0,  passiiu/ 
ichich  it  will  reach  +1°,  +2°,  etc.  In  this  illustration,  the  quantity  passes  from 
negative  to  positive  by  passing  through  0. 

It  appears  in  geometry,  that  a  quantity  may  also  change  its  sign  in  passing 
through  infinity.  Thus  the  tangent  of  an  arc  less  than  90°  is  positive  ;  but  if 
the  arc  continually  increases,  the  tangent  becomes  infinity  at  90°,  passing  which 
it  becomes  negative. 

Now,  as  we  know  of  no  other  way  in  which  a  varying  quantity  can  change  its 
sign,  it  is  assumed  as  a  fundamental  principle  in  mathematics  that,  if  a  vary- 
ing  QUANTITY  CHANGES   ITS   SIGN,   IT   PASSES   THROUGH   ZERO,  OR   INFINITY. 


NAMES  OF  DIFFERENT  FORMS  OF  EXPRESSION. 
54:»  A  I^olyiioinial  is  an  expression  composed  of  two  or  more 


10  LITERAL   ARITHMETIC. 

parts  connected  by  the  signs  plus  and  minus,  each  of  which  parts  is 
called  a  term. 

5o.  A  3Ionoinial  is  an  expression  consisting  of  one  term ;  a 
Binomial  has  two  terms;  a  Trinomial  has  three  terms,  etc. 

06,  A  Coefficient  of  a  term  is  that  factor  which  is  considered 
as  denoting  the  number  of  times  the  remainder  of  the  term  is  taken. 
The  numerical  factor,  or  the  product  of  the  known  factors  in  a  term, 
is  most  commonly  called  the  coefficient,  though  any  factor,  or  the 
product  of  any  number  of  factors  in  a  term  may  be  considered  as 
coefficient  to  the  other  part  of  the  term. 

tT7.  Similar  Terms  are  such  as  consist  of  the  same  letters 
affected  with  the  same  exponents. 


SECTION  IL 
ADDITION. 


58.  Addition,  is  the  process  of  combining  several  quantities,  so 
that  the  result  sliall  express  the  aggregate  value  in  the  fewest  terms 
consistent  with  the  notation. 

59*  The  Snm  or  Ainount  is  the  aggregate  value  of  several 
quantities,  expressed  in  the  fewest  terms  consistent  with  the  nota- 
tion. 

60,  JProp,  1,  Similar  terms  are  united  hy  Addition  into  one. 

Dem. — Let  it  be  required  to  add  4rt€,  hac,  —  2ac,  and  —  dac.  Now  4ac  is  4 
times  ac,  and  5ac  is  5  times  the  same  quantity  {ac).  But  4  times  and  5  times  the 
same  quantity  make  9  times  that  quantity.  Hence,  4ac  added  to  ryrtc  make  9ac. 
To  add  —  2ac  to  9ac  we  have  to  consider  that  the  negative  quantity,  —  2ac,  is  so 
opposed  in  its  character  to  the  positive,  9ac,  as  to  tend  to  destroy  it  when  com- 
bined (added)  with  it.  Therefore,  —  2ac  destroys  2  of  the  9  fmes  nc,  and  gives, 
when  added  to  it,  7ac.  In  like  manner,  —  Sac  added  to  Tr^r,  gives  4ac.  Thus  the 
four  similar  terms,  4nc,  tiac,  —  2ac,  and  —  Sac,  have  been  combined  (added)  into 
one  term,  4ac  ;  and  it  is  evident  that  any  other  group  of  similar  terms  can  be 
treated  in  tlie  same  manner.     <j,  E.  D. 

01,  Cor.  1. — In  adding  similar  termsy  if  the  terms  are  all  posi- 
tive, the  sum  is  positive ;  if  all  negative,  the  sum  is  negative ;  if 
some  are  positive  and  some  negative,  the  sum  takes  the  sign  of  that 
kind  (positive  or  negative)  loMch  is  in  excess. 

ScH. — The  operation  of  adding  positive  and  negative  quantities  may  look 
to  the  pupil  like  Subtraction.     For  example,  we  say  +5  and  —3  added  make 


i? 


ADDITION.  11 

-f  2.  This  looks  like  Subtraction,  and,  in  one  view,  it  is  Subtraction.  But 
why  call  it  Addition  ?  The  reason  is,  because  it  is  simply  putting  the  quanti- 
ties together — aggregating  them — not  finding  their  difference.  Thus,  if  one 
boy  pulls  on  his  sleigh  5  pounds  in  one  direction,  while  another  boy  pulls  3 
pounds  in  the  opposite  direction,  the  combined  (added)  effect  is  2  pounds  in 
the  direction  in  which  the  first  pulls.  If  we  call  the  direction  in  which  the 
first  pulls  positive,  and  the  opposite  direction  negative,  we  have  +5  and  —3 
to  add.  This  gives,  as  illustrated,  +2.  Hence  we  see,  that  the  sum  of  +5 
and  —3  is  +2. 

But  the  difference  of  +  5  and  —  3  is  8,  as  will  appear  from  the  following 
illustration :  Suppose  one  boy  is  trying  to  draw  a  sleigh  in  a  certain  direction, 
and  another  is  holding  back  3  lbs.  If  it  takes  10  lbs.  to  move  the  sleigh,  the 
first  boy  will  have  to  pull  13  lbs.  to  get  it  on.  But  if,  instead  oi  holding  hach 
3  lbs.,  the  second  boy  jpt^Aes  5  lbs.,  the  first  boy  will  have  to  pull  only  5  lbs. 
Thus  it  appears,  that  the  difference  between  pushing  5  lbs.  (or  +  5)  and  hold- 
ing back  3  lbs.  (—3)  is  8  lbs. 

In  like  manner  the  sum  of  $25  of  property  and  $15  of  debt,  that  is  the 
aggregate  value  when  they  are  combined,  is  $10.  +25  and  —15  are  +10. 
But  the  difference  between  having  $25  in  pocket,  and  being  $15  in  debt,  is 
$40.     The  difference  between  +25  and  —15  is  40. 

G^,  Cor.  2. — The  sum  of  two  qnantities,  the  one  positive  mid  the 
other  negative,  is  the  numerical  difference,  toith  the  sign  of  the  greater 
prefixed. 

OS,  Cor.  3.—/^  appears  that  addition  in  mathematics  does  not  al- 
ways imply  increase.  Whether  a  quantity  is  increased  or  diminished 
hy  adding  another  to  it,  depends  upon  the  relative  nature  of  the  two 
quantities.  If  they  both  tend  to  the  same  end,  the  result  is  an  increase 
in  that  direction.  If  they  tend  to  ojjposite  ends,  the  result  is  a  dimi- 
nution of  the  greater  by  the  less. 


64,  I^rop,  2,  Dissimilar  terms  are  not  united  into  07ie  by  addi- 
tion, but  the  operation  of  adding  is  expressed  by  writing  them  in 
succession,  with  the  positive  terms  preceded  by  the  +  sign,  and  the 
negative  by  the  —  sign. 

Dem. — Let  it  be  required  to  add  +  4<!y*,  +  3«&,  —  2icy,  and  —  mn.  ^cy^  is  4 
times  c«/',  and  3rf6  is  3  times  (th,  a  different  quantity  from  cy'^ ;  the  sum  will, 
therefore,  not  be  7  times,  nor,  so  far  as  we  can  tell,  any  number  of  times  ry^  or 
a}),  or  any  other  quantity,  and  we  can  only  exprcHS  the  addition  thus :  4cy^  i  3a6. 
In  like  manner,  to  add  to  this  Fum  —  S;?'//  we  can  only  (>xpress  the  addition,  as 
4ny2  -f  ^(ih  4-  (_  2.ry).  But  since  "2xy  is  nejjative,  it  tends  to  destroy  the  positive 
quantities  and  will  take  out  of  them  Ixy.  Hence  the  result  will  be  4c?/ '^  +  3«& 
—  2a;y.  The  effect  of  —  mn  will  be  the  same  in  kind  as  that  of  —  2.r^,  and 
hence   the  total    sum   will  be    \cy^  +  Zcih  —  2xy  —  mn.     As  a  similar  course 


12  LITERAL  ARITHMETIC. 

of  reasoning  can   be   applied   to  any   case,  the  truth   of   the   proposition   ap' 
pears. 

Sen. — In  such  an  expression  as  4cy '  +  dab  —  %vy  —  w;?,the  —  sign  before  the 
mil  does  not  signify  that  it  is  to  be  taken  from  the  immediately  preceding 
quantity  ;  nor  is  this  the  signification  of  any  of  the  signs.  But  the  quan- 
tities having  the  —  sign  are  considered  as  operating  to  destroy  any  which 
may  have  the  +  sign,  and  vice  versa. 

Go,  Cor. — Adding  a  negative  quantity  is  the  same  as  subtracting 
a  numerically  equal  positive  quantity  ;  that  is,m  +  (—  n)  is  m  —  w, 
shown  as  above. 

Dem. — Since  a  negative  quantity  is  one  which  tends  to  destroy  a  positive 
quantity,  —  ii  when  added  to  /«  (t.  €.  +  m)  destroys  n  of  the  units  in  m,  and 
lience  gives  as  a  result  m  —  n. 


60,  I^rob, — To  add  polynomials, 

RULE.  —  Combine  each  set  of    similar  terms   into   one 

TERM,  AND  CON'NECT  THR   RESULTS   WITH   THEIR  OWN   SIGNS.        ThE 
polynomial  THUS   FOUND    IS  THE   SUM   SOUGHT.* 

Dem. — The  purpose  of  addition  being  to  combine  the  quantities  so  as  to 
express  the  aggregate  (sum)  in  the  fewest  terms  consistent  with  the  notation, 
the  correctness  of  the  rule  is  evident,  as  only  similar  terms  can  be  united  into 
one  (60,  64). 


67,  Prop,  »?,  Literal  terms,  which  arc  similar  only  toith  respect 
to  part  of  their  factors,  may  be  united  into  one  term  ivith  a  polynomial 
coefficient. 

Dem. — Let  it  be  required  to  add  5ax,  —  2cx,  and  2mx.  These  terms  are 
similar,  only  with  respect  to  x,  and  we  may  say  5a  times  x  and  —  2c  times  x 
make  {5a  —  2c)  times  x,  or  (5a  —  2c)x.  And  then,  5a  —  2c  times  x  and  2m  times 
X  make  (o«  —  2c  +  2m)  times  x,  or  (5a  —  2c  +  2m)x.    q.  e.  d. 


68,  Prop,  4,  Compound  terms  which  have  a  common  compound, 
or  polynomial  factor,  may  be  regarded  as  similar  and  added  with 
respect  to  that  factor. 

Dem.  5{x^  —  y^),  2{:X-  —  y-)  and  —  3(j;^  —  y^)  make,  when  added  with  re- 
spect to  {x^  —  y-),  4(.r-  —  y-),  for  they  are  5  +  2  —  3,  or  4  times  the  same  quan- 
tity {x-  —  y^).     In  a  similar  manner  we  may  reason  on  other  cases.     Q.  E.  D. 

♦  This  is  the  proficient's  rule,  as  exhibited  on  p^ge  45  of  the  Complete  School  Algebra, 
ScH.  2. 


ADDITION.  13 

ScH. — The  object  and  process  of  addition,  as  now  explained,  will  be 
seen  to  be  identical  witli  the  same  as  the  pupil  has  learned  them  in  Arith- 
metic, except  what  grows  out  of  the  notation,  and  the  consideration  of 
positive  and  negative  quantities.  For  example,  in  the  decimal  notation  let 
it  be  required  to  add  248,  10506,  5003,  81,  and  106.  The  units  in  the  several 
numbers  are  similar  terms,  and  hence  are  combined  into  one  :  so  also  of  the 
tens,  and  of  the  liundreds.  The  process  of  carrying  has  no  analogy  in  the 
literal  notation,  since  the  relative  values  of  the  terms  are  not  supposed  to  be 
known.  Again,  there  is  nothing  usually  found  in  the  decimal  addition  like 
positive  and  negative  quantities.  With  these  two  exceptions  the  processes 
are  essentially  the  same.  The  same  may  be  said  of  addition  of  compound 
numbers. 


Examples. 

1.  Find  the  sum  of  2a  —3x^,  bx^  —  la,  —  3«  +  z^,  and  a  —  Sx^, 

2.  Find  the  sum  of  a^  -  b^  -h  Sa^b  -  5ab^,   3a^  -  4a^b  +  Sb^ 

-  3ab\  a^  +b^  +  daH,  2a^  -  U^  -  5ab^,  6aH  +  lOab^,  and  -  (Ja^ 

-  la^b  +  ^ab^  +  2b^. 

3.  Find   the   sum   of  bca^x^  +  Ua^x^  +  mx^y^,   and   lOca'-x^ 

—  2ba^x^  -f  67nx^y^. 

4.  Add  2x^  —  4:X^  +  x^,  bx^y  —  ab  +  x^,  4x^  -  x^,  and  2xi  -  3 

5.  Add  ^(x  +  y)  and  |(.t  —  y), 

6.  Add  ax-\-2by  +  cz,  ^x  +  Vy  +  Vz,  Sy^'-2xi  +  Sx^,  4:cz  -  Sax 

—  2by,  and  2ax  -  Wy  -  2A 

7.  Add  cz  —  2ay,  2az  —  Say,  my  —  az,  with  respect  to  z  and  y. 

8.  Add  {a-\-b)Vx-(2-hifi)\/y,^y^  +  (a  +  c)x^,3uVy—{2d-e)x^, 
—2nVx  +  12a  Vy,  and  (m  +  n)ij^-}-(b  +  2c)\/x. 

9.  Add  x^  +  xy  ■¥  y^,  ax^  —  axy  +  ay^,  and  —  by^  +  bxy  +  bx^, 

10.  Add  a(x  +  y)  +  b(x  —  y),  m{x  i-  y)  —  n{x  —  y). 

11.  Add  'SmVx  —  y  +  QnVx  —  y  —  ^^x  -—  y  —  Sn^Jx  —  y. 

12.  Add  Sax'i  -f  hy~^  —  2c,  — = 1-  8c,  and  —  ^ax~^  —  my^ 

Vx    y 

-de. 

13.  Add  i^/a^  -  x^,  -^^/^^^l^,  and  ^a^  -  x^. 


14  LITERAL   ARITHMETIC. 

14.  Add  J^-tA.-^,  iL_-_iJ_^,  and  -  2a{x^  -  1)4. 

Vx^  -  1     {x-  -  ip 

15.  Add  i(l/^2  -  y-i  +  2;-2),  and  |(a;t  +  -^-  -  :^). 

16.  Add    (a  -  b  +  c)y/x^~Y^y     {a  +  h  -  c)  {x^  ~  ?/2)i      and 
(b-{-c-  a)  Vx^  -  yK 

17.  Condense  tlie  polynomial  ^ax^  —  3?/«  +  %cz  —  4:mVx  i  3)ny^ 
^2ax^  +  Gcz,  into  2(a  —  2?7i)Vx  +3(;?i  —  l)^^  +  Scz. 


SECT/ON  IIL 
SUBTRACTION. 


CO.  Subtraction  is,  primarily,  the  process  of  taking  a  less 
quantity  from  a  greater.  In  an  enlarged  sense,  it  comes  to  mean 
taking  one  quantity  from  another,  irrespective  of  their  magnitudes. 
It  also  comprehends  all  processes  of  finding  the  difference  between 
quantities.  In  all  cases  the  result  is  to  be  expressed  in  the  fewest 
terms  consistent  with  the  notation  used. 

70,  TJie  Difference  between  two  quantities  is,  in  its  primary 
signification,  the  number  of  units  which  lie  between  them ;  or,  it  is 
what  must  be  added  to  one  in  order  to  pi'odttce  the  other.  When  it  is 
required  to  take  one  quantity  from  another,  the  difference  is  what 
must  be  added  to  the  Subtrahend  in  order  to  produce  the  Minuend, 

71.  JProb, — To  perform  Subtraction, 

RULE. — Change  the  signs  of  each  term  in  the  subtra- 
hend FROM  +  TO  — ,  OR  FROM  —  TO  +,  OR  CONCEIVE  THEM  TO 
BE   CHANGED,    AND   ADD   THE   RESULT  TO   THE   MINUEND. 

Dem. — Since  the  difference  sought  is  what  must  be  added  to  the  subtrahend 
to  produce  tlie  minuend,  we  may  consider  this  difference  as  made  up  of  two 
parts,  one  the  subtrahend  with  its  signs  changed,  and  the  other  the  minuend. 
When  the  sum  of  these  two  parts  is  added  to  the  subtrahend,  it  is  evident  that 
the  first  part  will  destroy  the  subtrahend,  and  the  other  part,  or  minuend,  will 
be  the  sum. 


SUBTRACTION.  15 


Thus,  to  perform  the  example  : 
From  5ax  —  6^  —  3(f  —  4m 

Take  2ax  +  2b  —  5d  +  Sm^ 


Subtrahend  with  signs  changed,     —  2ax  —  2b  -{-  5d  —  Sm 
Minuend,  5ax  —  &>  —  Sd  —  4m 


If  these  three 
quantities  are 
added  together, 
the     sum     will 

Difference,  Sax  —  8&  +  2d— 12m    evidently  be  the 

minuend.  If,  therefore,  we  add  the  second  and  third  of  them  (that  is,  the  sub- 
trahend, with  its  signs  changed,  and  the  minuend)  together,  the  sum  will  be 
what  is  necessary  to  be  added  to  the  subtrahend  to  produce  the  minuend,  and 
hence  is  the  difference  sought.     Q.  E.  D. 

72,  Con.  1. —  When  a  parenthesis,  or  any  symbol  of  like  significa- 
tion {4:4),  occurs  in  a  polynomial,  preceded  by  a  —  sign,  and  the 
parenthesis  or  equivalent  symbol  is  removed,  the  signs  of  all  the  terms 
which  were  within  must  be  changed,  since  the  —  sign  indicates  that 
the  quantity  tvithin  the  parenthesis  is  a  subtrahend. 

73,  Cor.  2. — A7iy  quantity  can  be  placed  ivithin  a  parenthesis, 
preceded  by  the  —  sign,  by  changing  all  the  signs.  The  reason  of 
this  is  evident,  since  by  removing  the  parenthesis  according  to  the 
preceding  corollary,  the  expression  loould  return  to  its  original  form. 


Examples. 

1.  How  much  must  be  added  to  8  to  produce  12  ?  What  is  the 
difference  between  8  and  12  ?  How  much  must  be  added  to  ^ax^ 
—  5^3  (the  subtrahend)  to  produce  ?>ax^  +  2^^  ? 

Aiistuer. — ^To  Zax^  we  must  add  bax^ ;  and  to  —  5^'  we  must 
add  +  7;y3.     Hence  in  all  we  must  add  hax^  +  '^y^- 

2.  From  3x^  -  2x^  -  x  -  7  take  2x^  -  3x^  +  x  +  1. 

3.  From  a^  —  x^  take  a^  +  2ax  -f  x^. 

4.  From  1  +  Sx^  +  3x  +  x^  take  1  -  Sx^  ^-Sx-  x^. 

5.  From  x'^  +  2x^y'^  +  y^  take  x'^  —  2x'^y^  +  y^. 

6.  From  7\/l  +  x^  -  3ay^  take  -  sVl  +  x^  +  Say^. 

7.  From  ay^  +  10 Vab  take  ay  +  x\/7ib. 

8.  From  bx^  —  3\/m?i  +  1  take   b^x  +  (mn)^  —  1. 

9.  From  a  +  b  -h  Va  —  b  take  b  -\-  a  —  (a  —  b)^  +  VaK 


16  LITERAL  ARITHMETIC. 

10.  Remove  the  parentheses  from  the  following: 

a-  {{b-c)  -  d]  ;         la-  j3a  -  [4a-  (5«  -  "Za)]]  ; 
^a  -  b)  -  c  +  d  -  {a  -  b  —'Z  (c  -  d)]  ; 
Z(U--b-c)-b  {a-  (U  +  c)\  +  3  \b-(c-a)\. 

11.  Include  within  brackets  the  3d,  4th,  and  5th  terms  of  Zab 
—  x^-hax  —  lOby  +  50.     Also  the  4th  and  5th.    Also  the  2d  and  3d. 

Theory  of  Subtraction. — Subtraction  is  finding  the  difference  between 
quantities,  that  is,  finding  what  must  be  added  to  one  quantity  to  produce  the 
other.  This  difference  may  always  be  considered  as  consisting  of  two  parts, one 
of  which  destroys  the  subtraliend,  and  the  other  part  is  the  minuend  itself. 
Hence,  to  perform  subtraction,  we  change  the  signs  of  the  subtrahend  to  get 
that  part  of  the  difference  which  dertroys  the  subtrahend,  and  add  this  result  to 
the  minuend,  which  is  the  other  part  of  the  difference. 

^«» 


SECT/ON  IV, 

MULTIPLICATION. 

74.  Multiplication  is  the  process  of  finding  the  simplest  ex- 
pression consistent  with  the  notation  nsed,  for  a  quantity  which 
shall  be  as  many  times  a  specified  quantity,  or  such  a  part  of  that 
quantity,  as  is  represented  by  a  specified  number. 

7i>.  CoK.  1. — The  mtdtlplier  must  alioays  be  conceived  as  an  ab- 
stract number^  since  it  shoivs  how  many  times  the  multiplicand  is 
to  be  taken. 

76,  Cor.  2. — The  product  is  always  of  the  same  hind  as  the  mul- 
tiplicand. 

77,  Prop,  1, — Tlie  product  of  several  factors  is  the  same  in 
whatever  order  they  are  taken. 

Dem. — 1st.  a  -K  &,  is  a  taken  6  times,  0Ta-\-a  +  a  +  a  +  a to  6  terms. 

Now,  if  we  take  1  unit  from  each  term  (each  a),  we  shall  get  &  units ;  and  this 
process  can  be  repeated  a  times,  giving  a  times  b,  or  b  x  a.     .'.  a  x  b  =  b  x  a. 

2d.  Wlien  there  are  more  than  two  factors,  as  abc.  We  have  shown  that  ab 
=  ba.  Now  call  this  product  in,  whence  ftbc  —  mc.  But  by  part  1st,  mc  =  cm. 
.'.  abc  =  bac  =  cah  =  cba.  In  like  manner  we  may  show  that  the  product  of  any 
number  of  factors  is  the  same  in  Avhatever  order  they  are  taken,    q.  e.  d. 


78,  JProp,  2, —  When  two  factors  have  the  same  sign  their  prod- 
uct is  positive:  when  they  have  different  signs  their  product  is  neg- 
ative. 


MULTIPLICATION.  17 

Dem. — 1st.  Let  tlie  factors  be  +  a  and  +  b.  Considering  a  as  the  multiplier 
we  are  to  take  +  b,  a  times,  which  gives  +  ab,  a  being  considered  as  abstract  in 
the  operation,  and  the  product,  +  «6,  being  of  the  same 'kind  as  the  multipli- 
cand ;  that  is,  positive.  Now,  when  the  product,  +  ab,  is  taken  in  connection 
with  other  quantities,  the  sign  +  of  the  multiplier,  a,  shows  that  it  is  to  be 
added ;  that  is,  written  with  its  sign  unchanged.     .'.  {+  b)  x  {+  a)  =  +  ab. 

2d.  Let  the  factors  he  —  a  and  —  b.  Considering  a  as  the  multiplier,  we  are 
to  take  —  b,a  times,  which  gives  —  «&,  a  being  considered  as  abstract  in  the 
operation,  and  the  product,  —  ab,  being  of  the  same  kind  as  the  multiplicand ; 
that  is,  negative.  Now,  when  this  product,  —  ab,  is  taken  in  connection  with 
other  quantities,  the  sign  —  of  the  multiplier  shows  that  it  is  to  be  subtracted ; 
that  is,  written  with  its  sign  changed.     :.  {—b)  x  {—  a)  =  +  ab. 

3d.  Let  the  factors  be  —  «  and  +  &.  Considering  a  as  the  multiplier,  we  are 
to  take  ■{■  b,  a  times,  which  gives  +  ab,  a  being  considered  as  abstract  in  the 
operation,  and  the  product,  ■{-  ab,heing  ot  the  same  kind  as  the  multiplicand; 
that  is,  positive.  Now,  when  this  product,  +  ab,  is  taken  in  connection  with 
other  quantities,  the  sign  —  of  the  multiplier  shows  that  it  is  to  be  subtracted ; 
that  is,  written  with  its  sign  changed.     .•.(+&)  x  {—  a)  =  —  ab. 

4th.  Let  the  factors  he  +  a  and  —  b.  Considering  a  as  the  multiplier,  we  are 
to  take  —  b,  a  times,  which  gives  —  ab,  a  being  considered  as  abstract  in  the 
operation,  and  the  product,  —  ab,  being  of  the  same  kind  as  the  multiplicand ; 
that  is,  negative.  Now,  when  this  product,  —  ab,  is  taken  in  connection  with 
other  quantities,  the  sign  +  of  the  multiplier  shows  that  it  is  to  be  added ;  that 
is,  written  with  its  own  sign.     .*.  {—  b)  x  {+  a)  =  —  ab.     q.  e.  d. 

70,  Cor.  1. —  The  j^^'oduct  of  any  riumher  of  positive  factors  is 
positive. 

SO,  Cor  2. — The  product  of  an  even  number  of  negative  factors  is 
positive. 

81,  Cor.  3. — TJie  product  of  an  odd  number  of  negative  factors  is 
7iegative. 


82,  Prop,  3, — Tlie  product  of  two  or  more  factors  consisting  of 
the  same  quantity  affected  with  exponents,  is  the  common  quantity 
with  an  exponent  equal  to  the  snni  of  the  exponents  of  the  factors. 
That  is  a'"  X  a"  =  «'""^" ;  or  rt'"-  a"-  a*  =  «"*+"+*,  etc.,  whether  the  expo- 
nents are  integral  or  fractional,  positive  or  negative. 

Dem. — 1st.   Wlien  the  exponents  are  positive  integers.    Let  it  be  required  to 

multiply  a"*  by  a""  and  a\    a'^  =  aaaa to  m  factors,  a"  =  aaaaa io  n 

factors,  and  a''  —  aaaaa to  a  factors.     Hence  the  i)roduct,  being  composed 

of  all  the  factors  in  the  quantities  to  be  multiplied  together,  contains  m  +  n  A-  s 
factors  each  flr,  and  hence  is  expressed  «'"+ "  +  *.  Since  it  is  evident  that  this  rea- 
soning can  be  extended  to  any  number  of  factors,  as  «"•  x  a"  x  a"  x  a"",  etc., 
etc.,  the  proposition  in  this  case  is  proved. 

2 


18  LITERAL  ARITHMETIC. 

2d.  WJien  the  exponents  are  positive  fractions.    Let  it  be  required  to  multiply 

mem 

a"  by  a* .     Now  a"  means  m  of  the  ii  equal  factors  into  which  a  is  conceived  to 
be  resolved.     If  each  of  these  n  factors  be  resolved  into  b  factors,  a  will  be  re- 

m 

solved  into  bn  factors.     Then,  since  a"  contains  m  of  the  n  equal  factors  of  a, 
and  each  of  these  is  resolved  into  b  factors,  m  factors  will  contain  bm  of  the  bn 

m  bm  c 

equal  factors  of  a.    Hence  a"  =  «'"' .     In  like  manner  a^  may  be  shown  equal  to 

en  m  c  t>m  en 

a''*  ;  and  a*  x  a^  =  a^'^  x  a^".    This  now  signifies  that  a  is  to  be  resolved  into 

m  e  ^ 

bn  factors,  and  bm  +  en  of  them  taken  to  form  the  product.     /.  a"  x  a^  =  a*"* 


xa*"  =  a    *"    ,ora"    *,  which  proves  the  proposition  for  positive  fractional 
exponents,  since  the  same  reasoning  can  be  extended  to  any  number  of  factors, 

m  e  e 

as  o"  X  a'*  X  a'',  etc. 

3d.  Whe7i  the  exponents  are  negative.    Let  it  be  required  to  multiply  «-"'  by 
a—",  m  and  w  being  either  integral  or  fractional.    By  definition  a—"^  x  «— *  = 

—  X  — .    Now,  as  fractions  are  multiplied  by  multiplying  numerators  together 
a**      a* 

and  denominators  together,  we  have  —  x  —  =  -—7-  by  part  1st  of  the  demon- 
^  a"»      a"       «"•  +  " 

stration.    But  this  is  the  same  as  a- (»«  +  ")  or  a-"'-".    .'.  a—""  x  a-"  =  a-"'-'* 


Examples. 

1.  Prove  as  above  that  81^  X  81*  =  81^'"  and  that  81^  =  81^ 

2.  Prove  that  m"  X  vi""  =  7)1"+". 

3.  Prove  that  16"*  x  16"*  =  16"*. 

4.  Prove  that  25"*  X  25*  is  1. 

5.  Prove  that  a~'  X  a^  is  a. 

ScH.— The  student  must  be  careful  to  notice  the  difference  between  the 
signification  of  a  fraction  used  as  an  exponent,  and  its  common  signification. 
Thus  I  tised  as  an  exponent  signifies  that  a  number  is  resolved  into  3  equal 
factors,  and  the  product  of  2  of  them  taken  •,  whereas  I  used  as  a  common 
fraction  signifies  that  a  quantity  is  to  be  separated  into  3  equal  parts,  and 
the  sum  of  two  of  them  taken. 

S3,  Proh, — To  muUiiyly  monomiaU. 

RULE. — Multiply  the  numerical  coefficiej^ts   as  in  the 

DECIMAL  NOTATION,  AND  TO  THIS  PRODUCT  AFFIX  THE  LETTERS  OF 
ALL  THE  FACTORS,  AFFECTING  EACH  WITH  AN  EXPONENT  EQUAL  TO 
THE    SUM    OF    ALL    THE    EXPONENTS    OF    THAT    LETTER    IN    ALL    THE 


MULTIPLICATION.  19 

FACTORS.  The  SIGN^  of  the  product  will  be  +  EXCEPT  WHEN^ 
THERE  IS  AN  ODD  NUMBER  OF  NEGATIVE  FACTORS;  IN  WHICH  CASE 
IT  WILL   BE   — . 

Dem. — This  rule  is  but  an  application  of  the  preceding  principles.  Since  the 
product  is  composed  of  all  the  factors  of  the  given  factors,  and  the  order  of  ar- 
rangement of  the  factors  in  the  product  does  not  affect  its  value,  we  can  write 
the  product,  putting  the  continued  product  of  the  numerical  factors  first,  and 
then  grouping  the  literal  factors,  so  that  like  letters  shall  come  together. 
Finally,  performing  the  operations  indicated,  by  multiplying  the  numerical 
factors  as  in  the  decimal  notation,  and  the  like  literal  factors  by  adding  the  ex- 
ponents, the  product  is  completed. 


84,  I^rob, — To  multiply  two  factors  together  token  one  or  hoth 
are  polynomials, 

R  ULE. — Multiply  each  term  of  the  multiplicand  by  each 

TERM  OF  THE  MULTIPLIER,  AND  ADD  THE  PRODUCTS. 

Dem. — Thus,  if  any  quantity  is  to  be  multiplied  by  a  -|-  &  —  c,  if  we  take  it  a 
times  {i.  e.  multiply  by  a),  then  h  times,  and  add  the  results,  we  have  taken  it 
a  +  h  times.  But  this  is  taking  it  c  too  many  times,  as  the  multiplier  required 
it  to  be  taken  a  +  h  minus  c  times.  Hence  we  must  multiply  by  e,  and  subtract 
this  product  from  the  sum  of  tlie  other  two.  Now  to  subtract  this  product  is 
simply  to  add  it  with  its  signs  changed  {71).  But,  regarding  the  —  sign  of  c 
as  we  multiply,  will  change  the  signs  of  the  product,  and  we  can  add  the  partial 
products  as  they  stand,  even  without   first   adding  the  products  by  a  and  h. 

Q.  E.  D. 


S5,  Theo. — TJie  square  of  the  sum  of  two  quantities  is  equal  to 
the  square  of  the  first^  plus  twice  the  product  of  the  two,  plus  the 
square  of  the  second. 

80,  Theo. — The  square  of  the  difference  of  two  quantities  is 
equal  to  the  square  of  the  first,  ininus  twice  the  product  of  the  two, 
plus  the  square  of  the  second. 

87,  Theo. — The  product  of  the  sum  and  difference  of  two  quan- 
tities is  equal  to  the  difference  of  their  squares. 

The  demonstration  of  these  three  theorems  consists  in  multiplying 
2^  +  y  by  ic  +  ^,  a;  —  ^  by  ic  —  y,  and  x  +  y  hy  x  —  y. 


20  UTERAL  AlilTHMETIG. 

Examples. 

1.  Multiply  together  Sax,  —  Sa^x^,  4:hij,  —  y^,  and  2x*y*. 

2.  Multiply  together  3x*,  —  mx^,  2m',  x-%  —  2,  and  2a;"*. 

3.  Multiply   together  40a;*,   x^y  and   f  A/^ ;    also   Sa^h^,   and 

«  _i  _I  i 

4.  Multiply  m^  by  m  ^,  a~*  by  a",  aH'""  by  a^ft*,  m  "  by  w*", 

V'rt  by  ^^,  ^^3  by  ^^. 

5.  Multiply  '6a  —  2hhy  a  +  4i. 

6.  Multiply  a;2  +  a;y  +  y^  by  a;*  —  xy  -{■  y^. 

7.  Multiply  7/1*  +  ?i*  +  0*  -  7)1^71^  —  m^o^  —  n^o^  by  m^  +  n» 

8.  Multiply  a"'  —  «"  +  rt^  by  ^T  —  a. 

9.  Multiply  together  z  —  a,  z  —  b,  z  —  c,  z  —  d. 

10.  Multiply  together  x  +  y,  x  —  y,  x^  -\-  xy  +  y^  and  a;*  —  xy 

SuG. — Try  the  factors  in  different  orders,  and  compare  the  labor  required. 

m        t  m  t_  nt.        t 

11.  Multiply  a^b'  *  -  cl^f  ''  +  1  by  a^'"^  +  1. 

12.  Multiply  2a^-''b^-''  4-  Sa^-^b""  by  10a''-^+^Z'"  +  »  -  oa'-'b-"^. 

13.  Square  the  following  by  the  theorems  (8S^  86)  : 

1+a,   x-2,    3/ +3^,    a~i-a~^b^,   a:"  +  a:,    f±-,     a;-*  +  .V, 
}rt^  -  i^r^'     Z/.c-i//~  n    —  ay-^xX    2a2J-(3-p)  _^  Ja^^r*. 

14.  Write  the  following  products  by  (87) : 

(1  +  frt)  X  (1  -  f«),     (99ax  4-  9a/«^)  X  (99rta:  -  da^x^). 

15.  Expand  (a  +  ^  +  c)   (a  +  b  —  c)  {a  — b  +  c)  (— a  +  b  +  c). 


MULTIPLICATION.  21 

MULTIPLICATIOIS'   BY   DeTACHED    COEFFICIElsrTS. 

88m  lu  cases  in  which  the  terms  of  both  multiplicand  and  multi- 
plier contain  the  same  letters,  and  can  be  so  arranged  that  the  ex- 
ponents of  the  same  letters  shall  vary  in  the  successive  terms  of 
each  according  to  the  same  law,  a  simiUir  laAV  ^vill  liold  good  in  the 
product,  and  the  multiplication  can  be  efifected  by  using  the  co- 
efficients alone,  in  the  first  instance,  and  then  writing  the  literal 
factors  in  the  product  according  to  the  observed  law.  A  few 
examples  will  make  this  clear : 

1.  Multiply  2a^  -  3a^x  +  ^ax^  —  x^  by  2a^  —  ax  +  7x^. 

OPERATION. 

2  -  3  +    5-    1 

2-1+7 


4  -  6  +  10  -    2 
-2+3-5+1 

+  14-21  +  35-7 


4-8  +  27-28  +  36 


Prod.,     4«6  -  8a^x  +  27a^x^  -  2Sa^x^  +  SQax^  -  7a:« 
2.  Multiply  a:3  +  2a;  —  4  by  rc2  —  1. 

SuG. — By  writing  these  polynomials  thus,  x^  +  Ox^  +  2x  —  4,  x^  +  Ox  —  1, 
the  law  of  the  exponents  in  each  case  becomes  evident.     Hence  we  have, 

1  +  0  +  2-4 
1  +  0-1 


1+0+2-4 

_l_0-2  +4 

1+0+1-4-2+4 

Prod.,      a;'  +  Oa;*  +  x^  —4x^  —  2x  +  4,  or  x'^  +  x^  —  4x^  —  2a;  +  4 

3.  Multiply  3«3  -j.  4:ax  -  ox^  by  2a^  -  6ax  +  4x^. 

4.  Multiply  2^3  -  3ah^  +  5h^  by  2a^  -  6b^. 

Bug.— The  detached  coefficients  are  2  +  0  —  3  +  5,  and  2  +  0  —  5. 

5.  Multiply  «3.  -\-aix  +  ax^  ■}■  x^  hy  a  —  x. 

6.  Multiply  x^  -  'dx^  +  3a;  -  1  by  x^  -  2x  -h  1. 


22  LITERAL  ARITHMETIC. 


SECTION  F. 

DIVISION. 

SO,  T>ivision  is  the  process  of  finding  how  many  times  one 
quantity  is  contained  in  another. 

00,  The  problem  of  division  maybe  stated:  Given  the  product 
of  two  factors  and  one  of  the  factors^  to  find  the  other  ;  and  the  siiffi- 
cient  reason  for  any  quotient  is,  that  midtiplied  by  the  divisor  it 
gives  the  dividend. 

01,  Cor.  1. — Dividend  and  divisor  may  both  be  multiplied  or 
both  be  divided  by  the  same  number  without  affecting  the  quotient. 

02,  Cor.  2. — If  the  dividend  be  multiplied  or  divided  by  any 
number,  while  the  divisor  remains  the  same,  the  quotient  is  multiplied 
or  divided  by  the  same. 

03,  Cor.  3. — If  the  divisor  be  multiplied  by  any  number  while  the 
dividend  remains  the  same,  the  quotient  is  divided  by  that  number  / 
but  if  the  divisor  be  divided,  the  quotient  is  midtiplied. 

94:,  Cor.  4. —  The  sum  of  the  quotients  of  two  or  more  quantities 
divided  by  a  common  divisor,  is  the  same  as  the  quotient  of  the  sum 
of  the  quantities  divided  by  the  same  divisor. 

05,  Cor.  5. —  The  difference  of  the  quotients  of  two  quantitie.\ 
divided  by  a  common  divisor,  is  the  same  as  the  quotient  of  the  dif- 
ference divided  by  the  same  divisor. 

These  corollaries  are  direct  consequences  of  the  definition,  and  need  no 
demonstration  ;   but  they  should  be  amply  illustrated. 

96,  Def. — Cancellation,  is  the  striking  out  of  a  factor  common  to  both 
dividend  and  divisor,  and  does  not  affect  the  quotient,  as  appears  from  {01), 


97,  Lemma  1. —  When  the  dividend  is  positive,  the  quotient  has 
the  same  sign  as  the  divisor  ;  but  when  the  dividend  is  negative,  the 
quotient  has  an  o2yposite  sign  to  the  divisor. 

08,  Lemma  2. —  When  the  dividend  and  divisor  consist  of  the 
same  quantity  affected  by  exponents,  the  quotient  is  the  common 
quantity  with  an  exponent  equal  to  the  exponent  in  the  dividend, 
m.inus  that  in  the  divisor. 


DIVISION.  23 

These  lemmas  are  immediate  cousequences  of  the  law  of  the  signs  and 
exponents  in  multiplication. 

99,  Cor.  1. — An^/  quantity  rcith  an  exponent  0  is  1,  since  it  may 
be  considered  as  arising  from  dividing  a  quantity  by  itself. 

Thus,  X  representing  any  quantity,  and  m  any  exponent,  a;"'  -5-  a;"*  =:  a;°  =  1. 

100,  Cor.  2. — Negative  exponents  arise  from  division  whe^i 
there  are  more  factors  of  any  number  in  the  divisor  than  in  the  divi- 
dend. 

101,  Cor.  3. — A  factor  may  be  transferred  from  dividend  to 
divisor  (or  from  numerator  to  denominator  of  a  fraction^  ichich  is 
the  same  thing),  and  vice  versa,  by  changing  the  sign  of  its  expo?ient. 


102 •  J^rob,  1, — To  divide  one  monomicd  by  another, 

RULE. — Divide  the  numerical  coefficient  of  the  divi- 
dend BY  THAT  OF  THE  DIVISOR  AND  TO  THE  QUOTIENT  ANNEX  THE 
LITERAL  FACTORS,  AFFECTING  EACH  WITH  AN  EXPONENT  EQUAL  TO 
ITS  EXPONENT  IN  THE  DIVIDEND  MINUS  THAT  IN  THE  DIVISOR,  AND 
SUPPRESSING  ALL  FACTORS  WHOSE  EXPONENTS  ARE  0.  ThE  SIGN 
OF  THE  QUOTIENT  WILL  BE  +  WHEN  DIVIDEND  AND  DIVISOR  HAVE 
LIKE   SIGNS,  AND  —  WHEN  THEY   HAVE   UNLIKE   SIGNS. 

Dem. — The  dividend  being  the  product  of  divisor  and  quotient,  contains  all 
the  factors  of  both ;  hence  the  quotient  consists  of  all  the  factors  which  are 
found  in  the  dividend  and  not  in  the  divisor. 


103,  Fvoh,  2, —  To  divide  a  j^olynomial  by  a  monomial. 
RULE. — Divide  each  term  of  the  polynomial  dividend  by 

THE   MONOMIAL   DIVISOR,  AND  WRITE  THE   RESULTS   IN  CONNECTION 
WITH   THEIR   OWN   SIGNS. 

Dem. — This  rule  is  simply  an  application  of  the  corollaries  {94,  95), 


104,  Dep. — A  polynomial  is  said  to  be  arranged  with  reference  to  a  certain 
letter  when  the  term  containing  the  highest  exponent  of  that  letter  is  placed  first 
at  the  left  or  right,  the  term  containing  the  next  highest  exponent  next,  etc.,  etc. 


24  LITERAL  ARITHMETIC. 

105,  Prob.  S, — To  perform  division  when  both  dividend  an^ 
divisor  are  polynomials. 

RULE. — Having    arranged    dividend    and    divisor    with 

REFERENCE  TO  THE  SAME  LETTER,  DIVIDE  THE  FIRST  TERM  OF  THE 
dividend  by  the  first  TERM  OF  THE  DIVISOR  FOR  THE  FIRST 
TERM  OF  THE  QUOTIENT.  ThEX  SUBTRACT  FROM  THE  DIVIDEND 
THE  PRODUCT  OF  THE  DIVISOR  INTO  THIS  TERM  OF  THE  QUOTIENT, 
AND  BRING  DOWN  AS  MANY  TERMS  TO  THE  REMAINDER  AS  MAY 
BE  NECESSARY  TO  FORM  A  NEW  DIVIDEND.  DiVIDE  AS  BEFORF, 
AND   CONTINUE   THE    PROCESS   TILL    THE   WORK   IS   COMPLETE. 

Dem. — The  arrangement  of  dividend  and  divisor  according  to  the  same  letter 
enables  us  to  find  the  term  in  the  quotient  containing  the  highest  (or  lowest  if 
we  put  the  lowest  power  of  the  letter  first  in  our  arrangement)  power  of  the 
same  letter,  and  so  on  for  each  succeeding  term. 

The  other  steps  of  the  process  are  founded  on  the  principle,  that  the  product 
of  the  divisor  into  the  several  parts  of  the  quotient  is  equal  to  the  dividend. 
Now  by  the  operation,  the  product  of  the  divisor  into  the  Jird  term  of  the 
quotient  is  subtracted  from  the  dividend  ;  then  the  product  of  the  divisor  into  the 
second  term  of  the  quotient  ;  and  so  on,  till  the  product  of  the  divieor  into  each 
term  of  the  quotient,  that  is,  the  product  of  the  divisor  into  the  \cholc  quotient, 
is  taken  from  the  dividend.  If  there  is  no  remainder,  it  is  evident  that  this 
product  is  equal  to  the  dividend.  If  there  w  a  remainder,  the  product  of  the 
divisor  and  quotient  is  equal  to  the  whole  of  the  dividend  except  the  remainder. 
And  this  remainder  is  not  included  in  the  parts  subtracted  from  the  dividend,  by 
operating  according  to  the  rule. 

ScH. — Tliis  process  of  division  is  strictly  analogous  to  "  Long  Division  " 
in  common  arithmetic.  The  arrangement  of  the  terms  corresponds  to  the 
regular  order  of  succession  of  the  thousands,  hundreds,  tens,  units,  etc., 
while  the  other  processes  are  precisely  the  same  in  both. 

Examples. 

3  1       ^»  - 

1.  Divide  m^  by  w^,  /i"  by  n'^y  (ab)^'^  by  (ab)"  ,  a^  hy  a^,  a~* 

by  «5,  ;c  3  by  x'^,  x'^  by  x~^, 

„    ^        «-2^2     2ar^x~^y  ,   bcd-^bx-^     ^ 

2.  Free  -37—,,    »    o   -1    o-   and  -5-3^ — —-   from  negative  expo- 

nents,  and  explain  the  process. 

3.  Divide  15ay«  by  3ay,  Sa^b^a^d  by  ^aH^c^,  3ah^  by  a^b^, 
—  doa*'bx^hy1a^bx,-20aJbh  by  ~-4.0ab^c,  y" -by  y",  -?/  by  y-^ 
na^b^-'y  by  -^a-'b'-Py-",  -^a^b'^c^  by  -  12a'^bc^-'',  a'-'+^b'-'c^ 

m  n 

by  a*-^+i^''+Y'2,  and  xi>^j/~^  by  a;«y"i. 


DIVISION   BY   DETACHED   COEFFICIENTS.  7^ 

L  Divide  Ha^k-  -  l'2a^k^ +3a'k^  by  dak,  Ux^y^a^b  +  Ulx^y^ 
~  -^ii^ifab-    by    Wx'^y^,  Ibax^  —  Iba^x -\- 'dax   by    —  bax,   i:a'^^ni^ 

-  12ri-i»//i8  4- 5280   by  -  12«-i%    '^Q^x^y'"' -  2Vixy'''+'    by   l^xy, 
y^  +  3^2/  -  2^^   by  yK    V^"  -  b'^'"  -  &'+'"-  b'^*"  by   b'%  ax^ 

-  2fla;"«    '  +  3«.r  by  «2.-'+\ 

5.  Divide  4a;2  -  28z^  +  4%2  by  2.t  -  '7y. 

6.  Divide  G2*  —  Idax^  +  ISa^T^  — 13<«3a:-5rt*  by  2a;2— 3aa:— a^. 

7.  Divide  a;^  4-  ?/3  _{_  3^^  _  1  by  ^  4.  1^  _  1. 

8.  Divide  ««^i2  _  54  by  ab^  —  2,  x  —  4:J  by  x^  —  2«^. 

9.  Divide  xy-  ahy  x^y^  —  a^,  243^'  +  1024  by  4  +  Sa. 

10.  Divide  ^8  -  1^^*  +  Uy'  -  fe^  -  W^  +  |  by  ^2  _  |  +  5. 

11.  Divide  1  +  2x^  —  7x^  -  16x^  by  1  +  2.?:  +  3x^  +  4rc3. 

12.  Divide  {x^  -  y^f  by  (x  -  «/)^  a^  +  ^,-3  by  «  -j-  b'K 

13.  Divide  ?/* i^J  !/ • 

14.  Divide  1  by  1  —  x^,  also  by  1  +  ^^,  1  +  x,  and  by  1  —  a;. 

15.  Divide  «'+"  +  a^b  +  fli"  +  b'+"  by  ft"  +  b\ 

16.  Divide     a'"'-'"'b^''c  -  a"^+'-'Z>'-V  +  a-^b-'c"^  4-   a""""  Z»'''+V'^ 

-  «"«+«»- '^,3^.">-i  +  jp+x^.n.+«-i  by  «-»^-^-'  +  ^6''^-'. 

17.  Divide  ?>i"'+'  +  a^w^i""  +  n??!""  +  aii'"'^''  by  m  +  n. 

18.  Divide  7/m(a:«  +l)  +  (w2  -\- m^)  (x^+x)  +  {n^ +2nm){x^ +x^) 
by  «a:2  4-  /?z:c  4-  n. 

19.  Divide  Ma;*  4-  2(h  -  k)x^  -  {h^  4-  4  -  h^)x^  4-  2  (7i  4-  h)x 
^  Ilk  by  /l'a:2  —  7i  4-  2a7. 

20.  Divide   x  +  y  -{-  z  —  3  \/xyz  by  a;"^  4-  «/^  4-  ;z^ 


Division  by  Detached  Coefficients, 

106,  Division  by  detached  coefficients  can  be  effected  in  the  same 
cases  as  multiplication  (88).  The  student  will  be  able  to  trace  the 
process  and  see  the  reason  ji'oni  an  exampje. 


26  LITERAL  ARITHMETIC. 

1.  Divide  10a*  —  '^la^x  +  ^^^x^  -  l%ax^  -  8a;*   by  2a«  -  Soaj 

OPERATION, 

2  -  3  +  4)  10  -  27  +  34  -  18  -  8 1  5      -6     -2 

10  -  15  +  20  I  5a^  —  Qax  —  2a;»     Qiwt, 


-12  + 

-12  + 

14- 

18- 

18 
24 



4  + 
4  + 

6- 
6- 

-8 
-8 

2.  Divide  re*  -  Zax^  -  %a^x^  +  18«3:c  -  8a*  by  x*  +  2ax  —  2aK 

3.  Divide  6a*  -  96  by  3a  -  6. 

SuG.— The  detached  coefficients  are  6  +  0  +  0  +  0  -  96  and  3  —  6. 

4.  Divide  3y^  +  3xy^  —  Ax^y  —4:X^  hy  x  -\-  y. 

5.  Divide  x'^  +  y'^  hj  x  -h  y ;  »ilso  re*  —  y*  by  a;^  —  y*. 


Synthetic  Division. 

107,  When  division  by  detaclied  coefficients  is  practicable,  as  in 
tlie  examples  in  the  last  article,  the  operation  may  be  very  much 
condensed  by  an  arrangement  of  terms  first  proposed  by  W.  G.  Hor- 
ner, Esq.,  of  Bath,  Eng.,  which  is  hence  called  Horner's  method  of 
synthetic  division.  A  careful  inspection  of  tlie  operatiok  under 
Ex.  1,  in  the  last  article,  will  acquaint  the  student  with  the  process. 

Explanation  of  Operation. — Arrange  the 
coefficients  of  the  divisor  in  a  vertical  column 
at  the  left  of  the  dividend,  changing  the  signs  of 
all  after  the  first.  Draw  a  line  underneath  the 
whole  under  which  to  write  the  coefficients  of 
5a'-^x-2x',  Quot.         the  quotient. 

The  first  coefficient  of  the  quotient  is  found 
evidently  by  dividing  the  first  of  the  dividend  by  the  first  of  the  divisor, 
and  in  this  case  is  5.  As  the  first  term  of  the  dividend  is  always  destroyed  by 
this  operation,  we  need  give  it  (10)  no  farther  consideration.  Now,  multiplying 
the  other  coefficients  after  the  first  (t.  e.  +  3  and  —  4)  icith  their  sig-ns  changed, 
by  5,  we  have  +  15  and  —  20,  which  are  to  be  added  (?)  to  —  27  and  +  34.  Hence 
we  write  the  former  under  the  latter.  The  first  term*  of  the  second  partial  divi- 
dend can  be  formed  mentally  by  adding  (?)  +  15  to  —  27,  and  the  next  term  of 
the  quotient  by  dividing  this  sum  (—  12)  by  2.     Hence  —  6  is  the  second  term  of 

*  Strictly,  the  "  coefficient  of;  "  but  tUis  form  is  asedfor  breyity. 


operation. 

2 

10-27  +  34-18-8 

+  3 

+  15-20  +  24  +  8 

-4 

-18-6 

5     -6     -2 

SYNTHETIC   DIVISION. 


27 


the  quotient.  (We  did  not  add  (?)  —  20  to  +  34,  because  there  is  more  to  be 
taken  in  before  the  first  term  of  the  next  partial  dividend  is  formed.) 

Having  found  the  second  term  of  the  quotient  (—  6),  we  multiply  the  terms 
of  the  divisor,  except  the  first,  (with  their  signs  changed)  by  —  6,  and  write  the 
results,  —  18  and  +  24,  under  the  third  and  fourth  of  the  dividend,  to  which 
they  are  to  be  added  (?).     Now  we  have  all  that  is  to  be  added"*  to  +34  (viz., 

—  20  and  —  18)  in  order  to  obtain  the  first  term  of  the  next  partial  dividend. 
Hence,  adding,  we  get  —  4.  which  divided  by  2  gives  —  2  as  the  next  term  of 
the  quotient.  Multiplying  all  the  terms  of  the  divisor  except  the  first,  as  before, 
we  have  —  6  and  +  8,  which  fall  under  —  18  and  —  8.     Now  adding  +  24  and 

—  6  to  —  18,  nothing  remains.  So  also  +8  —  8  =  0,  and  the  work  is  complete, 
as  far  as  the  coefficients  of  the  quotient  are  concerned. 

2.  Divide  x^  -  bx'^  +  15a;*  -  Ux^  +  21x^  -  13a;  +  5  by  aj*  -  2x^ 
+  4a;2  -  2a;  +  1. 


OPERATION. 


Quot, 


1 

+  2 
-4 
+  2 
-1 


1 

-5  +  15 

-24  + 

27- 

13  + 

5 

+  2- 

4 

+    2- 

1  + 

3- 

•5 

6 

+  12- 
+  10- 

6  + 
20 

10 

j_ 

-3  + 

_^ 

0 

0 

0 

^ 

a;'  -  3a;  +  5 


3.  Divide  4i/6  -  Uy^  +  60y*  -  SOy'  +  my^  —  24y  +  4  by  %y^ 

-4^  +  2. 

4.  Divide  x'^  -  y'^  hj  x  —  y ',  also  1  by  1  —  a;. 

5.  Will   a;  +  2  divide   a;*  +  2a;'  —  7a;'  -  20a;  4-  12   without  a  re- 
mainder?    Willa;-3? 

6.  Will  a;  +  3,  or  a;  —  3,  divide  a^  —  6a;*  —  16a;  +  21  without  a  re- 
mainder ?     Will  a;  +  7,  or  a;  —  7  ? 


♦  The  student  will  not  fail  to  eee  that  this  addition  is  equivalent  to  the  ordinary  subtraction 
since  the  signs  of  the  terms  have  been  changed. 


28  LITERAL  ARITHMETIC. 


CHAPTER  n. 

FACTORING. 


SECTION  I. 
FUNDAMENTAL  PROPOSITIONS. 

108,  The  Factors  of  a  uumber  are  those  numbers  wliich  mul- 
tiplied together  produce  it.  A  Factor  is,  therefore,  a  Divisor.  A 
Factor  is  also  frequently  called  a  measure,  a  term  ai'ising  in  Geome- 
try. 

109,  A  Common  Divisor  is  a  common  integral  factor  of 
two  or  more  numbers.  The  Greatest  Common  Divisor  of  two  or 
more  numbers  is  the  greatest  common  integral  factor,  or  the  product 
of  all  the  common  integral  factors.  Common  Measure  and  Com- 
mon Divisor  are  equivalent  terms. 

110,  A  Common  Multiple  of  two  or  more  numbers  is  an 
integral  number  which  contains  each  of  them  as  a  factor,  or  which 
is  divisible  by  each  of  them.  The  Least  Common  Multiple  of  two 
or  more  numbers  is  the  least  integral  number  which  is  divisible  by 
each  of  them. 

111,  A  Composite  Number  is  one  which  is  composed  of 
integral  factors  different  from  itself  and  unity. 

112,  A  Prime  N'umber  is  one  which  has  no  integral  factor 
other  than  itself  and  unity. 

lis,  Numbers  are  said  to  be  Prime  to  each  other  when  they  have 
no  common  integral  factor  other  than  unity. 

ScH.  1. — The  above  definitions  and  distinctions  have  come  into  use  from 
considering  Decimal  Numbers.  They  are  applicable  to  literal  numbers  only 
in  an  accommodated  sense.  Thus,  in  the  general  view  which  the  literal  no- 
tation requires,  all  numbers  are  composite  in  the  sense  that  they  can  be  fac- 


FACTORING.  29 

tored ;  but  as  to  whether  the  factors  are  greater  or  less  than  unity,  integral 
or  fractional,  we  cannot  affirm. 


114,  Prop,  1, — A  monomial  viay  be  resolved  into  literal  fac 
tors  by  separating  its  letters  hito  any  number  of  groups,  so  that  the 
sum  of  all  the  exponents  of  each  letter  shall  fnake  the  exponent  of 
that  letter  in  the  given  monomial. 


1 15,  Prop,  2, — Any  factor  which  occurs  in  every  term  of  a 
polynomial  can  be  removed  by  dividing  each  term  of  the  poly^iomial 
by  it. 

116,  Proj},  3, — If  two  terms  of  a  trinomial  are  positive  and 
the  third  ter)a  is  twice  the  jyroduct  of  the  square  roots  of  these  two, 
and  POSITIVE,  the  trinomial  is  the  square  of  the  SUM  of  these  square 
roots.  If  the  third  term  is  negative,  the  trinomial  is  the  square  of 
the  DIFFERENCE  of  the  two  roots. 


117,  Prop,  4, —  The  difference  between  two  quantities  is  equal 
to  the  product  of  the  sum  and  difference  of  their  square  roots. 


118,  Prop,  S, —  When  one  of  the  factors  of  a  quantity  is  givefi, 
to  find  the  other,  divide  the  given  quantity  by  the  given  factor,  and 
the  quotient  will  be  the  other. 


110,  Prop,  6, —  The  difference  between  any  two  quantities  is  a 
divisor  of  the  difference  between  the  same  powers  of  the  quan^ 
titles. 

The  SUM  of  two  quantities  is  a  divisor  of  the  difference  of  the 
same  EVEN  jyowers,  and  the  SUM  of  the  same  ODD  powers  of  the  quan- 
tities. 

DE\f. — Let  X  and  y  be  any  two  quantities  and  n  any  positive  integer.  First, 
x  —  y  divides  a;"  —  y".  Second,  if  n  is  even,  x  +  y  divides  a^  —  y".  Third,  if  n  is 
odd,  X  -\-  y  divides  ic"  +  y*. 


30  LITERAL  ARITHMETIC. 


FIRST. 

Taking  the  first  case,  we  proceed   in   form   with  the  division,  till  four 
of  the 

terms  of  the  x  —  y)a;"  —  y*       (a;"-'  +  a^-gy  +  a;"-^y2  +  g^-ys  +  etc, 

quotient  (enough  to  ^^^^-^^e^y_ 

determine  the  law)  are  x'^-^y  —  y" 

found.     We  find  that  each  x^  '^y  —  x^-^y^ 

remainder  consists  of  two  terms,  x'^-^y'^  —  y" 

the  second  of  which,  —  y",  is  the  x*-^y^  —  x'—'y^ 

second  term  of  the  dividend  constantly  a;"-  "y  3  _  y» 

brought  down  unchanged;  and  the  first  x^-^y^  —  x''-*y^ 

contains  x  with  an  exponent  decreasing  by  a?"-*y*— y* 

unity  in  each  successive  remainder,  and  y  with  an 

exponent  increasing  at  the  same  rate  that  the  exponent  of  x  decrecbses.  At  this 
rate  the  exponent  of  ar  in  the  nth  remainder  becomes  0,  and  that  of  y,  n.  Hence 
the  Tith  remainder  is  y"  —  y*  or  0  ;  and  the  division  is  exact. 

SECOND  AND  THIRD. 


X  +  y>r"  ±  y»         (a;"-'  -  ar"-«y  +  ^"-'y^  -  xr-*yi 

,etc. 

a?-  +  a--V 

a--*y2  ±  yn 
Taking  x  +  y                     ar^-'ys  +  a;"-'yS 

for  a  divisor,  we                                    —a^-^y^±y^ 

observe  that  the  exponent                 —  x^-'y^  —  x^'-^y^ 

of  x  in  the  successive  re-                                       a;"-*y4  ±  y" 

mainders  decreases,  and  that  of  y  increases 

the  same  as  before.  But  now  we  observe  that  the  first  term  of  the  remainder  is 
—  in  the  odd  remainders,  as  the  1st,  3d,  5th,  etc.,  and  +  in  the  ceen  ones,  as  the 
2d,  4th,  6th,  etc.  Hence  if  n  is  emn,  and  the  second  term  of  the  dividend  is  —  y", 
the  nth  remainder  is  y"*  —  y"  orO,  and  the  division  is  exact.  Again,  if  n  is  odd, 
and  the  second  term  of  the  dividend  is  +  y» ,  the  nth  remainder  is  —  y"  +  y" , 
or  0,  and  the  division  is  exact,     q.  e.  d. 

120,  Cor, —  The  last  proposition  applies  equally  to  cases  involv- 
ing fractional  or  7iegative  exponents. 

Dem. — Thus,  x^—y^  divides  x^—y^,  since  the  latter  is  the  difference  between 
the  4th  powers  of  x^  and  y*.  So  in  general  a;"  «♦  —  y  '  divides  x  »*  —  y  ^ ,  a 
being  any  positive  integer.  This  becomes  evident  by  putting  x  «i=v,  and 
y^r  —qff.  whence  x^'^  =  v',  and  y  ^  =  vf*.  But  ««  —  zo«  is  divisible  by  v~w, 
hence  x~  »  —  y  ~  is  divisible  by  a;    »  —  y    ' . 


FACTORING.  31 

121,  JProp,  7, — A.  trinomial  can  he  resolved  into  two  binomial 
factors,  when  one  of  its  terms  is  the  product  of  the  square  root  of 
one  of  the  other  two,  into  the  sum  of  the  factors  of  the  remaining  term. 
The  two  factors  are  respectively  the  algebraic  sum  of  this  square  root, 
and  each  of  the  factors  of  the  third  term. 

III. — Thus,  in  a;*  +  7aj  +  10,  we  notice  that  Ix  is  the  product  of  the  square 
root  of  x^,  and  2  +  5  (the  sum  of  the  factors  of  10).  The  factors  of  x-  +  Ix 
+  10  are  2;  +  2  and  x  4  5.  Again,  x^  —  ^  —  10,  has  for  its  factors  .t;  +  2  and 
«  —  5,  —  3.C  being  the  product  of  tlie  square  root  of  x-  (or  x),  and  the  sum  of 
—  5  and  2,  (or  —  3),  which  are  factors  of  -  10.  Still  again,  x^  +^x  —  10 
=  (a;  —  2)  {x  +  5),  determined  in  the  same  manner. 

Dem. — Tlie  trutli  of  this  proposition  appears  from  considering  the  product  of 
X  +  ahy  X  +  b,  which  is  x^  +  (a  +  b)  x  +  ab.  In  this  i)roduct,  considered  as  a 
trinomial,  we  notice  that  the  term  (a  +  b)x  is  the  product  of  fa;*  and  a  +  b,  the 
sum  of  the  factors  of  ab.  In  like  manner  (x  +  a)  (x  —  b)  z=x'  +  (a—  b)x  —  ab, 
and  (x  —  a)  {x  —  b)=x^  —  (a  +  b)x  ■{■  ah,  both  of  which  results  correspond  to  the 
enunciation.     Q.  E.  D. 

[Note. — In  application,  this  proposition  requires  the  solution  of  the  problem: 
Given  the  sum  and  product  of  two  numbers  to  find  the  numbers,  the  complete 
solution  of  which  cannot  be  given  at  this  stage  of  the  pupil's  progress.  It  Avill 
be  best  for  him  to  rely,  at  present,  simply  upon  inspection.] 


122,  I*i*op,  S, —  We  can  often  detect  a  factor  by  separating 
a  polynomial  hito  parts. 

Ex.  Factor  x^  +  12a;  -  28. 

Solution. — The  form  of  this  polynomial  suggests  that  there  may  be  a  bino- 
mial factor  in  it,  or  in  a  part  of  it.  Now  a;*  —  4c  +  4  is  the  square  of  a;  —  2, 
and  (.c«  -  4c  +  4)  +  (16a;-32)  makes  a;«  +  12aj  -  28.  But  (a;'-4r  +  4)  +  (16a;-32) 
=  {x-  2)  (a; -2)  +  (a;  -  2)16  =  (.i;  -  2)  (a;  -  2  +  16)  =  (a:  -  2)  (.c  +  14).  Whence 
X  —  %,  and  aj  +  14  are  seen  to  be  the  factors  of  x^  +  12a;  —  28. 


Miscellaneous  Examples. 

1.  Factor  Ifg^y  -  2Sf^gy^  4-  i2pgy,  ^x^y^  -  Hx^y^  +  UxyK 

2.  Factor  ?>?>  -  n^,  1  -  2V~v  +  x,  256«*  -f  544^2  +  289,  1  -  c\ 

3.  Factor  x^- x  -  TZ,y^- z^,a^ -^  b^,^  +  ^  _2,  a^ +23«  +  22. 

0^       a^ 

4.  Factor  ^  -  ---^  +  15.,  c«  -  d^,  c^  -  d-\  c^  -  d-\ 

m*       mx^      X*'  '  ' 


S2  LITERAL  ARITHMETIC. 


i        .J  ...4        K..-A    « 


5.  Factor  a'  -  m  %  4:t-*  —  5?/--*,  —  —  Ji  o,  a;S  4-  22a:  -  7623. 

6.  Factor  a;"  -  1,  507?n*  +  13267^2,^1  ^  867w3,  Vrt  -  V^. 

7.  Factor  x^-2ax  —  a^,  a""  dt  U^VcT^"  +  U^c"",  x^-\-\^+2J^. 

8.  Factor  A«*"  -  ^W"^'"+'  +  A^*"^',  3«  +  3^  -  61/^. 

9.  Resolve  x  into  two  equal  factors  ;  also  two  unequal  factors. 

10.  Resolve  dSx^y^z'^  —  3Vy*z  into  two  factors  of  which  one  is 
2y^Vz. 

11.  Resolve  121a^&^c^  into  two  equal  factors ;  also  into  four  equal 
factors. 

12.  Remove  the  factor  ^{ak^)^  from  S^a^k*. 

w*       7c-2  49rf» 

13.  Remove  the  factor  — ^  +  — ^^^om  m®w~*  —  ~K7r' 

14.  Remove  the  factor  a*  —  a^b  +  a^b^  —  aZ>^  4-  b^  from  «* 
+  b'. 

15.  Factor  15a  +  5rta;  —  a;  —  3,  21abccl—2%cdxy-{-\babmn—20mnxy, 
21a:2  +  232:^  -  20^2,  12^20;*  -  12rt2:2;}  +  3a«. 

16.  Factor  3.c3  -  12^3^2  _  4^2  +  1^  T2cd*m^  -  Ucdhn* 
+  9Gc2r^2;,j2, 

17.  The  terms  of  a  trinomial  are  ZOab,  9<t2  and  25^2,  What  sign 
must  be  given  to  each  that  the  trinomial  may  be  factored  ? 

18.  The  terms  of  a  trinomial  are  —  9rt,  \)i>^/a  and  4.  What  must 
be  the  signs  of  the  last  two  terms  that  the  trinomial  may  be 
factored  ? 

—  4  4- 

19.  Is  «  5  —  J'"  exactly  divisible  hy  a^  —  b  or  dJ  -\-  b  "i 

20.  Is  m^  —  n^  exactly  divisible  by  Vm  —  Vn't  by  Vm  +  Vn'i 
by  \/m  ±  ^7i  ? 

21.  Is  a;!*'!  +  y^^^  exactly  divisible  by  x  +  ?/  ?  by  a;  —  y  ? 

22.  Is  .-^2019  _|_  ^20  79  exactly  divisible  by  x'^  -  y"^  ?  by  .t^  +  y'^  ? 

23.  What  is  the  quotient  of  (%J  +  mz^)  -^  {k^   Vy  +  ^/m  z^)  ? 


HIGHEST  COMMON  DIVISOR.  38 

24.  What  is  the  quotient  of  {x^  +  y^)  -f-  (a^^V  +  ^tV)  ? 

25.  Write  the  following  quotients :  (a^  +  b^)  -^  (a^  +  ^2) ; 
(a;""  -  ;2'''*)  -^  (a;  -  z) ;  {x""  -  z'"")  -f-  (x  +  2;) ;  {x'"'+'  +  ^""+') 
-f-  (:r  +  2),  m  being  a  positive  integer. 

1  100 

26.  Factor  x^  +  ax  -{-  x  +  ay     1  —  a,     1  +  a,     -[^  —  —[-^    and 

x^  -X-  9900. 


^lU  yj 


27.  Factor    10rJ^+  |^]  -  20«,  4a;  +  4a;^  +  1  and  Sda'^  -  5b\ 

28.  x^  -x^  -2x  +  2,  6a;3  -  7«a;2  -  20a^Xy  x'"^  +  31af*  —  32. 


SECTION  IL 
GREATEST    OR    HIGHEST    COMMON    DIVISOR. 

123,  Def. — It  is  scarcely  proper  to  apply  the  term  Greatest  Common  Divisor 
to  literal  quantities,  for  the  values  of  the  letters  not  being  fixed,  or  specific, 
(jreat  or  small  cannot  be  affirmed  of  them.  Thus,  whether  « '  is  greater  than  a, 
depends  upon  whether  a  is  greater  or  less  than  1,  to  say  nothing  of  its  character 
as  positive  or  negative.  So,  also,  we  cannot  with  propriety  call  a^  —  y '  greater 
than  a  —  y.  If  a  =  i,  and  y  =:  \,  a^  —  y '  =  ^4,  and  a  —y  =  \  -^  .-.in  this  case 
^ '  —  y '  <  a  —  y.  Again,  if  a  and  y  are  both  greater  than  1,  but  a  <y,a^  —  y* 
though  numerkaUy  greater  than  a—y  is  absolutely  less,  since  it  is  a  greater 
negative. 

Instead  of  speaking  of  G,  C.  D.  in  case  of  literal  quantities,  wc  should  speak 
of  the  Highest  Common  Dicisor,  since  what  is  meant  is  the  divisor  which  is  of 
the  highest  degree  with  reference  to  the  letter  of  arrangement. 

[Note. — The  general  rule  for  finding  the  Greatest  or  Highest  Common 
Divisor  is  founded  upon  the  four  following  lemmas.] 

Jl^4,  Lemm.a.  1. —  The  Greatest  or  Highest  G.  J),  of  two  or  more 
numbers  is  the  product  of  their  common  prime  fa^ctors. 

Dem. — Since  a  factor  and  a  divisor  are  the  same  thing,  all  the  common  fac- 
tors  are  all  the  common  divisors.  And,  since  the  product  of  any  number  of  fac- 
tors of  a  number  is  a  divisor  of  that  number,  the  product  of  all  the  common  prime 
factors  of  two  or  more  numbers  is  a  common  diiisor  of  those  numbers.  More- 
over, this  product  is  the  Oreatest  or  Highest  C.  D ,  since  no  other  factor  can  be  in- 
troduced into  it  without  preventing  its  measuring  (dividing),  at  least,  one  of  tho 
given  numbers.     Q.  E.  D.  3 


84  LITERAL  ARITHMETIC. 

Examples. 

1.  Whai  is  the  G.  C.  D.  of  72,  84,  and  180  ? 

Solution. — Resolve  the  numbers  into  their  prime  factors,  and  take  the  pro- 
duct of  those  which  are  common  to  all. 

2.  Find  the  G.  0.  D.  of  48,  204,  and  228. 

3.  Find  the  G.  C.  D.  of  81,  123,  and  315. 

4.  Find  the  Highest  C.  D.  of  %x^yz^  and  Ihx^y. 

Solution. — Here  we  see  that  x,  x,  and  y  are  all  the  literal  factors  com- 
mon to  both ;  and  since  8  and  15  have  no  common  factor,  x  x.  x  x  y  is  the 
Highest  C.  D. 

5.  Find  the  H.  C.  D.  of  UkH^m^  and  ^()kHhn^n*. 

6.  Find  the  H.  C.  D.  of  SaHc,  18aH^,  and  2(jaHhm, 

7.  Find  the  H.  C.  D.  of  Hx^i/'^z^  and  4xy-^zr-  \ 

8.  Find  the  H.  C.  D.  of  ba^x^i/  —  lOax^y  +  bax^y  and   Za'^x^y 

-  ^x^yK 

9.  Find  the  H.  C.  D.  of  a:^  -  a:  -  12  and  x^  -  x^  -  ^x  +  9. 

Solution.— a;«  -  a?  -  12  =  («  -  4)  (a?  +  3)  {121).  x^-x^-^x  +  ^  =  x^{x  -  1) 

-  9(a;  -  1)  =  («*  -  9)  («  -  1)  =  («  -  3)  {x  +  3)  (a:  -  1).  Now  we  see  that  a;  +3  is 
a  common  divisor  of  the  two  polynomials,  and  since  it  is  the  only  divisor  com- 
mon to  both,  it  is  the  H.  C.  D. 

10.  Find  the  H.C.D.of  4:h^x^  -  Ub^x^  -f  Ub^x-U^  and  U^x^ 

-  8b^x*  -  4b*x  +  8^2. 

J2i>,  ScH. — The  difficulty  of  factoring  renders  this  process  impracticable 
in  many  cases.  There  is  a  more  general  method.  But,  in  order  to  demon- 
strate the  rule,  we  require  three  additional  lemmas. 

120.  Lemma  2.— A  poly?iGmial  of  the  form  Ax"  +  Bx"-' 
+  Cx""'-  -  -  -  Ex  -f  F,  which  has  no  common  factor  in  every 
term.,  has  ow  divisor  of  its  own  degree  except  itself 

Dem. — 1st.  Such  a  polynomial  cannot  have  one  factor  of  the  n\\\  degree — its 
own — with  reference  to  the  letter  of  arrangement,  and  another  which  contains 
the  letter  of  arrangement,  for  the  product  of  two  siicli  factors  would  be  of  a 
higher  (or  different)  degree  from  the  given  polynomial. 

2d,  It  cannot  have  a  factor  of  the  n\\\  degree  with  reference  to  the  letter  of 
arrangement,  and  another  factor  which  does  not  contain  that  letter,  for  this  last 
factor  would  appear  as  a  common  factor  in  every  term,  which  is  contrary  to  the 
hypothesis.     Q.  E.  D. 


HIGHEST   COMMON   DIVISOR.  35 

127 •  Lemma  3. — A  divisor  of  any  number  is  a  divisor  of  any 
multiple  of  that  number. 

III. — This  is  an  axiom.  If  a  goes  into  6,  q  times,  it  is  evident  that  it  goes 
into  n  times  &,  or  n6,  n  times  q,  or  nq  times. 

128,  Lemma  4. — A  common  divisor  of  tioo  numbers  is  a  divisor 
of  their  sum.  and  also  of  their  difference, 

Dem. — Let  a  be  a  C.  D.  of  m  and  n,  going  into  m,  p  times,  and  into  n,  q  times. 
Then  {m  ±  n)  -i-  a  =  p  ±  q.    Q.  E.  D. 

120.  I^rob, —  To  find  the  H.  C.  D.  of  two  polynomials  without 
the  necessity  of  resolving  them  hito  their  prime  factors. 

RULE. — 1st.  Arrangikg  the  polykomials  with  reference 
TO  the  same  letter,  and  uniting  into  single  terms  the  like 

POWERS  OF  THAT  LETTER,  REMOVE  ANY  COMMON  FACTOR  OR  FACTORS 
which  may  appear  in  all  the  TERMS  OF  BOTH  POLYNOMIALS,  RE- 
SERVING THEM  AS  FACTORS  OF  THE  H.  C.  D. 

2d.  Reject  from  each  polynomial  all  other  factors  which 

APPEAR  IN  EACH  TERM  OF  EITHER. 

3d.  Taking  the  polynomials,  thus  reduced,  divide  the  one 

WITH  the  greatest  EXPONENT  OF  THE  LETTER  OF  ARRANGEMENT, 
BY  THE  OTHER,  CONTINUING  THE  DIVISION  TILL  THE  EXPONENT  OF 
TH1E  LETTER  OF  ARRANGEMENT  IS  LESS  IN  THE  REMAINDER  THAN  IN 
THE  DIVISOR. 

4th.  Reject  any  factor  which  occurs  in  every  term  of  this 

REMAINDER,  AND  DIVIDE  THE  DIVISOR  BY  THE  REMAINDER  AS  THUS 
reduced,  treating  THE  REMAINDER  AND  LAST  DIVISOR  AS  THE 
FORMER  POLYNOMIALS  WERE.  CONTINUE  THIS  PROCESS  OF  REJECT- 
ING FACTORS  FROM  I:ACH  TERM  OF  THE  REMAINDER,  AND  DIVIDING 
THE  LAST  DIVISOR  BY  THE  LAST  REMAINDER  TILL  NOTHING  RE- 
MAINS. 

If,  at  any  TIME,  A  FRACTION  WOULD  OCCUR  IN  THE  QUOTIENT, 
MULTIPLY  THE  DIVIDEND  BY  ANY  NUMBER  WHICH  WILL  AVOID  THE 
FRACTION. 

The  LAST  DIVISOR  MULTIPLIED  BY  ALL  THE  FIRST  RESERVED  COM- 
MON FACTORS  OF  THE  GIVEN  POLYNOMIALS,  WILL  BE  THE  H.  C.  D. 
SOUGHT. 


3G  LITERAL  ARITHMETIC. 

Dem. — Let  A  and  B  represent  any  two  polynomials  whose  H,  C.  D.  is 
sought. 

1st.  Arranging  A  and  B  with  reference  to  the  same  letter,  for  convenience  in 
dividing,  and  also  to  render  common  factors  more  readily  discernible,  if  any 
common  factors  appear,  they  can  be  removed  and  reserved  as  factors  of  the  H. 
0.  D.,  since  the  H.  C.  D.  consists  of  all  the  common  factors  of  A  and  B. 

2d.  Having  removed  these  common  factors,  call  the  remaining  factors  C  and 
D.  We  are  now  to  ascertain  what  common  factors  there  are  in  C  and  D,  or  to 
find  their  H,  C.  D.  As  this  H.  C.  D.  consists  of  only  the  common  factors,  we  can 
reject  from  each  of  the  polynomials,  C  and  D,  any  factors  which  are  not  common. 
Having  done  this,  call  the  remaining  factors  E  and  F. 

3d.  Suppose  polynomial  E  to  be  of  lower  degree  with  respect  to  the  letter  of 
arrangement  than  F.  (If  E  and  F  are  of  the  same  degree,  it  is  immaterial  which 
is  made  the  divisor  in  the  subsequent  process.)  Now,  as  E  is  its  own  only  divisor 
of  ito  own  degree  (Lem.  2),  if  it  divides  F,  it  is  the  H,  C.  D.  of  the  two.  If,  in 
attempting  to  divide  F  by  E  to  ascertain  whether  it  is  a  divisor,  fractions  arise, 
F  can  be  multiplied  by  any  number  not  a  factor  in  E  (and  E  has  no  monomial 
factor),  since  the  common  factors  of  E  and  F  would  not  be  affected  by  the  opera- 
tion. Call  such  a  multiple  of  F,  if  necessary,  F'.  Then  the  H.  C.  D.  of  E  and 
F',  is  the  H.  C.  D.  of  E  and  F.  If,  now,  E  divides  F',  it  is  the  H.  C.  D.  of  E  and 
F.     Trying  it,  suppose  it  goes  Q  times,  with  a  remainder,  R. 

4th.  Any  divisor  of  E  and  F'  is  a  divisor  of  R,  since  F'  —  QE  =  R,  and  any 
divisor  of  a  number  divides  any  multiple  of  that  number  (Lem.  3),  and  a  divisor 
of  two  numbers  divides  their  difference.  The  H.  C.  D.  divides  E,  hence  it  di- 
vides QE,  and,  as  it  also  divides  F',  it  divides  the  difference  between  F'  and  QE, 
or  R.  Therefore  the  H.  C.  D.  of  E  and  F',  is  also  a  divisor  of  E  and  R,  and  can- 
not be  of  higher  degree  than  R. 

5th.  We  now  repeat  the  reasoning  of  the  3d  and  4th  paragraphs  concerning 
E  and  F,  with  reference  to  E  and  R.  Thus,  R  is  by  hypothesis  of-lower  degree 
than  E ;  hence,  dividing  E  by  it,  rejecting  any  factor  not  common  to  both,  or  in- 
troducing any  one  into  E,  which  may  be  necessary  to  avoid  fractions,  we  ascer- 
tain whether  R  is  a  divisor  of  E.  If  it  is,  it  divides  P',  since  F'  =  R  -f-  ^^  (Lem. 
8,  4),  and  hence  id  the  H.  C.  D.  of  E  and  F'. 

6th.  Proceeding  thus,  till  two  numbers  are  found,  one  of  which  divides  the 
other,  the  last  divisor  is  the  H.  C.  D.  of  E  and  F,  since  at  every  step  we  sliow 
that  the  H.  C.  D.  is  a  divisor  of  the  two  numbers  compared,  and  the  last  divisor 
is  its  own  H.  D. 

7th.  Finally,  we  have  thus  found  all  the  common  factors  of  A  and  B,  the  pro- 
duct of  which  is  their  H.  C.  D.     Q.  e.  d. 


Examples. 

1.  Find  the  H.  C.  D.  of  UaH^  +  U^y^  -  Ibah^y  +  12^2^^  -f-  Uy^ 

-  Ibahy"-,  and  QaH^  -  QaH^y  -  U^y'^  -^^ab^y^  +  QaHy  -  6««%« 

-  2hy*  4-  2aby\ 


HIGHEST  COMMON   DIVISOR.  37 

OPERATION. 

12a^b^  +  db'y^  -  l^db'-y  +  X'ia'hy  +  36y '  -  l^aby^      -    - {A). 

6ffl^6'  -  6a-6^y  -  26-.y »  +  2ah'y-  +  Qa'by  -  6g^-6y^  -  26y*  +  2<i6y^*    -    -  {B). 

4a-'&  +  6y-  —  Ort%  +  4«/'?/  +  y'  -  oay^ (C). 

^a^h  -  3a^&y  -  ?>y^^  +  ciby'-  +  Sa'y  -  'da'y'  -y'  +  ay' {D). 

(46  +  ^y)a-'  -  {5by  +  by') a  +  (6^=^  +  y') {E). 

(35  +  ^y)a'  -  (Sby  +  dy')a'  +  {by'  +  y')a-  {by^  +  y*) {F), 

(5)                                  W 
4a'  -  5ya  +  y-)  da^  -    3^^^  +    y'a  -     y=* 
^ 

(/)----    12(1'  -  12y^^  +    4y'a  -    4yX3a 

(70  -    -    -    -    12fl^'  -  Wya'  +    3y'a 

{L)    ------    -    'dya-  +      yVt  -    4y* 

£ 

(Jf) 12ya^  +    4y'a-Wy^{Sy 

(iV)  ------    -  12yf<^  -  15y'fl^ -H    by' 

(0)    -    -    Reject  19y'     -    -    -    19y=^a  -  19y '  (7 

(P)    -    - a  —  y)4a'—5ya  +  y^{4a-'i/, 

4yg  —  4yflg 

—  ya  +  y' 

:.  The  H.  C.  D.  of  (A)  and  (B)  is  (6)  (b  4-y)  {a  -  y)  =  ab'  +  aby  -  b^y  -  by^. 

ScH. — It  often  occurs  that  one  or  more  of  the  above  steps  are  not  required, 
especially  the  removing  of  a  compound  factor  from  the  given  iwlynomials. 

2.  Find  the  H.  C.  D.,  with  respect  to  ar,  of  x*  -  %x^  +  2\x^  -  20» 
-h  4,  and  2x^  -  \%x^  +  21a;  -  10. 

OPERATIOK. 

2ic=»  -  12«*  -I-  21ar  -  10)aj*  -    Sr'  +  21a?«  -  20tJ  +    4 


(C) -    2a!-'  -  lftc=*  +  42.1"'  -  40a;  -I-    8(« 

2a?''  -  12.g=^  4-  2U'*  -  lOx 

-  4»'»  -h  21^;''  -  3ac  -f-    8 

-  4g^  -f  24g^  -  42a;  +  20 
(D)  Reject  -  3 -    ac'*  +  12aj  -  12 


(^) 

cpi          I'P  A.     A. 

x'  - 

Ax  +  4)2.c^  -  nx'  +  2\x  -  10(2a;  -  4 

2^3  _    8,^2  +    8.C 

-    4a;''^  +  13a;  -  10 

-    4a;2  +  16a;  -  16 

-     Reject  -  3    -      -  3a;  +    6          {E) 
x-2)x'  -4x  +  i(x-2 

x^  -2x 

-2a; +  4 

-2a;  4-  4 

Hence  a;  -  2  is  the  H,  C.  D. 


38  LITERAL  ARITHMETIC. 

3.  Find  the  H.  C.  D.  of  2x^  +  5  -  8.r  +  x^,  and  42:c8  +  30  -  nx. 

4.  Find  the  H.  C.  D.  of  2ax^  +  2a  -{-  4a«,  and  7b  +  14^>.r  +  ^bx^ 
+  UbxK 

5.  Find  the  H.  C.  D.  of  Ga^  +  Hax  -  dx^,  and  6a^  +  llax  +  3x^. 

6.  Find   the    H.  C.  D.  of  4a3  _  4«2  -  ab*  +  b\  and   4^2  +  2ab 

7.  Find    the    H.  C.  D.  of    12a;*  -  2^x^ij  +  12x*y^,    and    Sx^t/^ 

-  24a;2^3  ^  24a;y*  -  8y«. 

8.  Find  the  H.  C.  D.  of  62ax^  -  2^ax*^  -  Uax*  -  12a  +  Sax^ 
+  GOax,    and  Ua^b   +  60a*bx*  -  16a*bx^   +  2rt«Z>a;«  -  74fl2^a; 

-  2aHxK 


ISO,  I^vob, —  To  find  the  H.  C.  D.  of  three  or  more  polynomials. 

RULE. — FiN'D  THE  H.  C.  D.  of  any  two  of  the  given  poly- 
nomials JJY  one  of  the  foregoing  methods,  and  then  find 
thk  H.  C.  D.  of  this  H.  C.  D.  and  one  of  the  remaining  poly- 
nomials, and  then  again  compare  this  last  H.  C.  D.  with 
another  of  the  polynomials,  and  find  their  H.  C.  D.  Con- 
tinue this  process  till  all  the  polynomials  have  been 
used. 

Dem. — For  brevity,  call  the  several  polynomials,  A,  B,  C,  D,  etc.  Let  the  H.  C. 
D.  of  A  and  B  be  represented  by  P,  whence  P  contains  all  the  factors  common 
to  A  and  B.  Finding  the  H.  C.  D.  of  P  and  C,  let  it  be  called  P'.  P',  therefore, 
contains  all  the  common  factors  of  P  and  C  ;  and  as  P  contains  all  that  are 
common  to  A  and  B,  P'  contains  all  that  are  common  to  A,  B,  and  C.  In  like 
manner  if  P"  is  the  H.  C.  D.  of  P'  and  D,  it  contains  all  the  common  factors  of 
A,  B,  C,  and  D,  etc.    Q.  E.  d. 

Examples. 

1.  Find  the  H.  C.  D.  of  x^  +  Ux  +  30,  2x^  +  21a;  +  54,  and  9a;3 
H-  h'^x*  -^x-  18.  The  H.  C.  D.  is  x  +  Q. 

2.  What  is  the  H.  C.  D.  of  lOa;^  +  10:^3^2  +  20a;*y,  2x^  +  2y8, 
and4y*  +  \2x^y^  +  ^x^y  +  12xy^  ? 


LOWEST    COMMON   MULTIPLE.  39 


SECTION  III. 

LOWEST  OR  LEAST  COMMON  MULTIPLE. 

131*  Def.  —  In  speaking  of  decimal  numbers,  the  term  Least  Common 
Multiple  is  correct,  but  not  in  speaking  of  literal  numbers.  For  example,  the 
numbers  («  +  h)'^  and  («*  —  h'^)  are  both  contained  in  {a  -f  h)'-  x  {a  —  h),  and  in 
any  multiple  of  this  product,  as  m{a  +  b)'  (a  —  h).  But  whether  7?i{a  +  by-  (a—h) 
is  greater  or  less  than  {a  +  b)'^  {a  —  b)  depends  upon  whether  a  is  greater  or  less 
than  b,  and  also  whether  m  is  greater  or  less  than  unity.  In  speaking  of  literal 
numbers,  we  should  say  Lowest  Common  Multiple,  meaning  the  multiple  of  low- 
est degree  with  respect  to  some  specified  letter. 


132,  Pvoh, — To  find  the  L.  C.  M,  of  two  or  more  numbers. 

RULE. — Take  the  literal  number  of  the  highest  degree, 
or  the  largest  decimal  number,  and  multiply  it  by  all  the 
factors  found  in  the  next  lower  which  are  not  in  it. 
Again,  multiply  this  product  by  all  the  factors  found  in 

THE  NEXT  LOWER  NUMBER  AND  NOT  IN  IT,  AND  SO  CONTINUE 
TILL  ALL  THE  NUMBERS  ARE  USED.  ThE  PRODUCT  THUS  OBTAINED 
IS  THE   L.  C.  M. 

Dem. — Let  A,  B,  C,  D,  etc.,  represent  any  numbers  arranged  in  the  order  of 
their  degrees,  or  values.  Now,  as  A  is  its  own  L.  M.,  the  L.  C.  M,  of  all  the 
numbers  must  contain  it  as  a  factor.  But,  in  order  to  contain  B,  the  L.  C.  M. 
must  contain  all  the  factors  of  B.  Hence,  if  there  are  any  factors  in  B  which  are 
not  found  in  A,  these  must  be  introduced.  So,  also,  if  C  contains  factors  not 
found  in  A  and  B,  they  must  be  introduced,  in  order  that  the  product  may  con- 
tain C,  etc.,  etc.  Now  it  is  evident  that  the  product  so  obtained,  is  the  L.  C.  M. 
of  the  several  numbers,  since  it  contains  all  the  factors  of  any  one  of  them,  and 
hence  can  be  divided  by  any  one  of  them,  and  if  any  factor  were  removed  it 
would  cease  to  be  a  multiple  of  some  one  or  more  of  the  numbers.     Q.  E.  d. 

1.  Find  the  L.  C.  M.  of  {x^  -  1),  (x'^  -  1),  and  (x  +  1). 

Solution. — The  L.  C.  M.  must  contain  a;'  —  1,  and  as  it  is  its  own  L.  M.,  if  it 
contains  all  the  factors  of  the  other  two,  it  is  the  required  L.  C'.  M.  The  factors 
of  a; '  —  1  are  (x  —  l){x^  +  x  +  1).  But  this  product  does  not  contain  the  factors 
of  («'  —  1).  which  are  {x  +  1)  {x  —  1).  Hence,  we  must  introduce  the  factor 
{x  +  1),  giving  (a;'  —  1)  (a;  -I-  1),  as  the  L.  C.  M.  of  .tr '  —  1  and  x-  —  1.  Now,  as 
this  product  contains  the  third  quantity,  it  is  the  L,  C.  M.  of  the  three, 

2.  Find  the  L.  CM.  of  (^+  by,a^  -b^,  {a-  b)^,3Lnda^  +  3a'-b 
+  dab^  -f  ^. 


^  LITERAL  ARITHMETIC. 

3.  Find  the  L.  C.  M.  of  (x^  -  4),  (x^  +  2),  and  (x^  -  2). 

4.  Find  the  L.  C.  M.  of  («*  -  2a^  +  1),  (1  +  a),  (a  -  1),  and  4. 

5.  Find  the  L.  C.  M.  oVda^b^xy,  57ax^,  87y3,  and  9a«6i 

G.  Find  the  L.  C.  M.  of  (1  -  18a  +  81^2),    (3«2  +  1)  (1  -  SVa), 

and  (27r7l-9rt  -  3a/«  +  1). 

Sen. — In  applying  this  rule,  if  the  common  factors  of  the  two  numbers  are 
not  readily  discerned,  apply  the  method  of  finding  the  H.  C.  D.,  in  order  to 
discover  them. 

7.  Find  the  L.  C.  M.  of  x^-2ax^  +  4a«a;  -  Sa^,  x^  +  2ax^+  ia^x 
+  8a 3,  and  x-  —  4a «. 

Solution. — The  L.  C.  M.  of  these  numbers  must  contain  a;'  —  2ax*  +  ^a^x 

—  8«^  ;  and  as  it  is  its  own  L.  M.,  if  it  contains  all  the  factors  of  x^  +  2ax^ 
+  Aa^x  +  8rt  ',  it  is  the  L.  C.  M.  of  thcsr  two  iK)lynomials.  But  as  the  common 
factors  of  these  numbers,  if  they  have  any,  are  not  readily  discerned,  we  apply 
the  nu'thod  of  II.  V.  D.,  and  find  that  x^  +  4a*  is  the  H.  C.  D.  of  the  two.  Since, 
then,.T'  —  '2  >  '  -  4// '  r  —  8a'  contains  the  factor  x^  +  4a*  of  the  second  number, 
it  is  only  lu  (  r-<aiy  to  introduce  the  other  factor  in  order  to  have  the  L.  C.  M.  of 
the  two.  ^'ow,  (.c '  +  2rt.i-'  +  4rt«.c  +  8a')  -h  {x^  +  4a*)=.r  +  2a.  Hence,  (.c='—2a.T* 
+  Aa^x—  8a')(.r  +  2a)  or  .r*  —  16a^  is  theL.  C.  M.  of  the  first  two  numbers, 
since  it  contains  all  the  factors  of  each,  and  no  more.  Now,  to  find  whether 
X*  —  16a  ^  is  a  multiple  of  the  remaining  number,  x^  —  4a',  or,  if  it  is  not,  what 
factors  must  be  introduced  to  make  it  so,  we  proceed  in  the  same  way  as  with 
the  first  two  numbers.  But  our  first  step  (or  117)  shows  us  that  x*  —16a''  is  a 
multiple  of  x^  —  4a'.     .'.  x*  —  16a*  is  the  L.  C.  M.  of  the  three  given  numbers. 

8.  Find  the  L.  C.  M.  of  x^  -  3x  -  70  and  x^  -  d9x  +  70. 

9.  Find  the  L.  C.  M.  o^  x^^  x  -  2,  x^-- x  -  6,  and  x^  -  4x  +  3. 

10.  Find  the  L.  C.  M.  of  a^-  AaH  +  9a^2_  io63  and  a^-\-  2aH 
-3a^>2-h  20^3. 

11.  Find   the  L.  C.  M.   of  x^-  ^x^  +  2^x  -  24,  x^-  Wx^  +  3lx 

—  30,  and  x^  -  Ux^  +  SSx  -  40. 

12.  Find  the  L.  CM.  of  a:*-10a;2  +  9,  rr* +10a;3 +20.T»-10a;~21, 
and  z*  +  4a;3  -  22a;2  —  4a;  +  21. 


FRACnONB.  41 


OHAPTEE  IIL 
jPJB^  cti on s. 


DEFINITIONS   AND  FUNDAMENTAL  PRINCIPLES. 

133.  A  Fraction,  in  the  literal  notation,  is  to  be  considered 
as  an  indicated  operation  in  Division. 

134.  ScH. — In  the  literal  notation  it  becomes  impracticable  to  consider 
the  denominator  as  indicating  the  number  of  equal  parts  into  which  unity  is 
divided,  and  the  numerator  as  indicating  the  number  of  those  parts  repre- 
sented by  the  fraction,  since  the  very  genius  of  this  notation  requires  that 

the  letters  be  not  restricted  in  their  signification.     Thus  in  -,  it  will  not  do 

to  say,  b  represents  the  number  of  equal  parts  into  which  unity  is  divided, 
since  the  notation  requires  that  whatever  conception  we  take  of  these 
quantities  should  be  sufficiently  comprehensive  to  include  all  values. 
Hence  h  may  be  a  mixed  number.  Now  suppose  ft  =  4|.  It  is  absurd  to 
speak  of  unity  as  divided  into  4}  equal  parts. 

135.  Cor.  1. — Since  numerator  is  dicidend  and  denominator 
divisor,  it  follows  from  (01^  f)2,  03)  that  dirndincf  or  midtiply- 
ing  both  terms  of  a  fraction  does  not  alter  its  value  ;  that  midti- 
plying  or  dividing  the  numerator  multiplies  or  divides  the  value  of 
the  fraction ;  and  that  multiplying  or  dividing  the  denominator 
divides  or  multiplies  the  fraction. 

136.  Cor.  2. — A  fraction  is  midtiplied  by  its  denominator  by 
simply  removing  it. 

137.  The  terms  Integer  or  Entire  Number,  Mixed  Number, 
Proper  and  Improper,  are  applied  to  literal  numbers,  but  not  with 
strict  propriety. 

Whether  m  +  n  is  an  integer,  a  mixed  number,  or  a  fraction,  depends  upon 
the  values  of  m  and  n,  which  the  genius  of  the  literal  notation  requires  to  be 
understood  as  perfectly  general,  until  some  restriction  is  imposed. 

As  a  matter  of  convenience,  we  adopt  the  following  definitions  : 


42  LITERAL  ARITHMETIC. 

13S,  A  number  not  having  the  fractional /or»i  is  said  to  have 
the  Integral  Form ;  as  m  +  n,  2c^d  —  Sa'^x  +  Sx^y*. 

139.  A  polynomial  having  part  of  its  terms  in  the  fractional 
and  part  in  the  integral  form,  is  called  a  Mixed  Wiunber, 

140.  A  Proper  Fraction,  in  the  literal  notation,  is  an  ex- 
pression wholly  in  the  fractional  form,  and  which  cannot  be  expressed 
in  the  integral  form  without  negative  exponents. 

By  calling  such  an  expression  a  proper  fraction,  we  do  not  assert  anything 

a 
with  reference  to  its  value  as  compared  with  unity.    Thus  -r-  is  a  proper  frac- 
tion, though  it  may  be  greater  or  less  than  unity.     It  may  also  be  written 

141.  An  Improper  Fraction  is  an  expression  in  the  frac- 
tional form,  but  which  can  be  expressed  in  the  integral  or  mixed 
form  without  the  use  of  negative  exponents. 

142.  A  Simple  Fraction  is  a  single  fraction  with  both 
terms  in  the  integral  form. 

143.  A  Compound  Fraction  is  two  or  more  fractions  con- 
nected by  the  word  of. 

This  term  is  not  generally  applicable  in  the  literal  notation.    Thus  we  may 

3         3  a     .  m        ,  ,  ^         .  , 

write  -^  of  ^  with  propriety,  but  not  y  of  — ,  unless  a  and  h  are  mtegral,  so 

that  the  fraction  -j-  may  be  considered  as  representing  equal  parts  of  unity,  as  ^ 

does.     If  the  word  of  is  considered  as  simply  an  equivalent  for  x  ,  the  notation 
is  of  course,  always  admissible.     But  it  is  scarcely  a  simple  equivalent. 

144.  A  Complex  Fraction  is  a  fraction  having  in  one  or 
both  its  terms  an  expression  of  the  fractional  form. 

145.  A  fraction  is  in  its  Lowest  Terms  when  there  is  no  com- 
mon integral  factor  in  both  its  terms. 

146.  I7ie  Lowest  Comtnon  Denominator  is  the  num- 
ber of  lowest  degree,  which  can  form  the  denominator  of  several 
given  fractions,  giving  fractions  of  the  same  values  respectively, 
while  the  numerators  retain  the  integral  form. 

147.  Heduction,  in  mathematics,  is  changing  the  form  of  an 
expression  without  changing  its  value. 


FRACTIONS.  43 

Reductions. 
14:8.  There  are  five  principal  reductions  required  in  operating 
with  fractions,  viz. :  To  Lowest  Terms,— From  Improper  Fractions 
to  Integral  or  Mixed  Forms,— Front  Integral  or  Mixed  Forms  to  hn- 
iwoper  Fractions, — To  Forms  haimig  a  Common  Denominator,— 
and  from  the  Complex  to  the  Simple  Form.. 

14:9  •  JProb.  1. —  To  reduce  a  fraction  to  its  lowest  terms. 
RULE. — Reject  all  common^  factors  from  both  terms;  or 

DIVIDE    both   terms    BY   THEIR    H.  C.  D. 

Dem. — Since  the  numerator  is  the  dividend  and  the  denominator  the  divisor, 
rejecting  tlie  same  factors  from  each  does  not  alter  the  value  of  the  fraction 
{fH).  Having  rejected  all  the  common  factors,  or,  what  is  the  same  thing,  the 
H.  C.  D.  (which  contains  all  the  common  factors),  the  fraction  is  in  its  lowest 
terms  {145). 

ScH.  1. — Since  the  H.  C.  D.  is  the  product  of  all  the  common  factors 
(109),  the  above  process  is  equivalent  to  dividing  both  terms  of  the  frac- 
tion by  their  H.  C.  D.  Whenever  the  common  factors  of  the  terms  are  not 
readily  discernible,  the  process  for  finding  their  H.  C.  D.  (129)  may  be 
resorted  to. 

ScH.  2. — The  opposite  process  is  sometimes  serviceable,  viz.:  the  intro- 
duction of  a  factor  into  both  terms  of  a  fraction,  which  will  give  it  a  more 
convenient  form.  It  requires  no  special  ingenuity  to  solve  such  problems, 
since,  if  the  factor  does  not  readily  appear,  it  can  be  found  by  dividing  a 
term  of  one  fraction  by  the  corresponding  term  of  the  other. 


ISO,  Prob,  2, —  To  reduce  a  fraction  from  an  improper  to  an 
integral  or  mixed  form. 

RULE.— Ferfoiui  the  division"  indicated  {133), 

1S1»    Cor. — Bg   means   of   negative   indices    {expo7ie7its)    any 
fraction  can  he  expressed  in  the  integral  form. 


IS 2,  Pvob,  3, — To  reduce  numbers  from  the  integral  or  mixed 
to  the  fractional  form. 

RULE. — Multiply  the  integral  part  by  the  given  de- 
nominator, and  annexing  the  numerator  of  the  frac- 
tional part,  if  any,  write  the  sum  over  the  given  de- 
nominator. 


44  LITERAL  ARITHMETIC. 

Dem. — In  the  case  of  a  number  in  the  integral  form,  the  process  consists  of 
multiplying  the  given  number  by  the  given  denominator  and  indicating  the 
division  of  the  product  by  the  same  number,  and  hence  is  equivalent  to  multi- 
plying and  dividing  by  the  same  quantity,  which  does  not  change  the  value  of 
the  number.  The  same  is  true  as  far  as  relates  to  the  integral  part  of  a  mixed 
form,  after  which  the  two  fractional  parts  are  to  be  added  together.  As  they 
have  the  same  divisors,  the  dividends  can  be  added  upon  the  principle  that  the 
sum  of  the  quotients  equals  the  quotient  of  the  sum  {94). 


IS 3.  Prob,  4,— To  reduce  fraction  having  differetU  denomi- 
nators  to  equivalent  fractions  having  a  common  denominator. 

RULE. — Multiply   both  terms  of  each  fraction^  by  the 

DENOMINATORS   OF   ALL  THE  OTHER   FRACTIONS. 

Dem. — This  gives  a  common  denominator,  because  each  denominator  is  the 
product  of  all  the  denominators  of  the  several  fractions.  The  value  of  any  one 
of  the  fractions  is  not  changed,  because  both  numerator  and  denominator  are 
multiplied  by  the  same  number  {135). 

lo4,  CoR. —  To  reduce  fractions  to  equivalent  ones  having  the 
Lowest  Common  Denominator.^  find  the  L.  C.  M.  of  all  the  denomi- 
nators for  the  new  denominator.  Then  multiply  both  terms  of  each 
fraction  by  the  quotient  of  that  L.  C.  M.  divided  by  the  denomhiator 
of  that  fraction. 


ISS,  Pvob,  S, — To  reduce  complex  fractions  to  the  form  of 
simple  fractions. 

RULE. — Multiply  numerator  and  denominator  of  the  com- 
plex FRACTION  BY  THE  PRODUCT  OF  ALL  THE  DENOMINATORS  OF 
the  partial  FRACTIONS  FOUND  IN  THEM;  OR,  MULTIPLY  BY  THE 
L.  C.  M.  OF  THE  DENOMINATORS  OF  THE  PARTIAL  FRACTIONS.* 

Dem. — This  process  removes  the  partial  denominators,  since  each  fraction  is 
multiplied  by  its  own  denominator,  at  least,  and  this  is  done  by  dropping  the 
denominator.  It  does  not  alter  the  value  of  the  fraction,  since  it  is  multiplying 
dividend  and  divisor  by  the  same  quantity. 


Addition. 
ISO,  Pvob. —  To  add  fractions. 

R  ULE. — Reduce  them  to  forms  having  a  common  denomina- 
tor, if  they  have  not  such  forms,  and  then  add  the  numera- 
tors, AND  write  the  SUM  OVER  THE  COMMON  DENOMINATOR. 

*  The  pnpil  Is  snppoped  to  have  obtained  sufficient  knowledge  of  fractions  in  common  arith- 
metic to  perform  these  operations. 


FRACTIONS.  4^ 

Dem. — The  reduction  of  the  several  fractions  to  forms  having  a  common  denomi- 
nator, if  they  have  not  such  forms,  does  not  alter  their  values  {135),  and  hence 
does  not  alter  the  sum.  Then,  when  they  have  a  common  denominator  (divisor), 
the  sum  of  the  several  quotients  is  equal  to  the  quotient  of  the  sum  of  the  sev- 
eral dividends  divided  by  the  common  divisor,  or  denominator  (f>i^). 

1S7»  Cor. — Expressio7is  in  the  mixed  form  may  either  be  reduced 
to  the  improper  form  and  then  added,  or  the  integral  parts  may  be 
added  into  one  sum,  and  the  fractional  into  another,  and  these  restdts 
added. 


SUBTRACTIOK. 

IS 8,  JPvoh, — To  subtract  fractions. 

RULE. — Reduce  the  fraction's  to  forms  having  a  common 

DENOMINATOR,  IF  THEY  HAVE  NOT  SUCH  FORMS,  AND  SUBTRACT  THE 
numerator  of  the  subtrahend  FROM  THE  NUMERATOR  OF  THE 
MINUEND,  AND  PLACE  THE  REMAINDER  OVER  THE  COMMON  DENOMI- 
NATOR. 

Dem, — The  value  of  the  fractions  not  being  altered  by  the  reduction,  their  dif- 
ference is  not  altered.  After  this  reduction,  we  have  the  difference  of  two  quo- 
tients arising  from  dividing  two  numbers  (the  numerators)  by  the  same  divisor 
(the  common  denominator).  But  this  is  the  same  as  the  quotient  arising  from 
dividing  the  difference  between  the  numbers  by  the  common  divisor  {95). 

ISO,  Cor. — Mixed  nmnbers  may  be  subtracted  by  annexing  the 
mbtrahend  with  its  signs  changed,  to  the  minuend,  and  then  combining 
the  term^  as  much  as  may  be  desired.  The  reason  for  the  change  of 
signs  is  the  same  as  in  whole  numbers  (71)- 


Multiplication. 

160,  I^rob.  1, —  To  midtiply  a  fraction  by  an  integer. 

E ULE.— M.\JLTITLY  THE  NUMERATOR  Oil  DIVIDE  THE  DENOMI- 
NATOR. 

Dem. — Since  numerator  is  dividend  and  denominator  divisor,  and  the  value  of 
the  fraction  is  the  quotient,  this  rule  is  a  direct  consequence  of  {92,  93). 

161,  JProb,  2, —  To  multiply  by  a  fraction. 

RULE. — Multiply  by  the  numerator  and  divide  by  the 

DENOMINATOR.* 

*  It  is  assumed  that  the  pupil  knows  how  to  divide  a  fraction  by  an  integer,  from  his  study 
of  arithmetic.  Nevertheless  the  problem  will  b«  introdaced  hereafter  for  the  purpose  of  famil- 
iarizing the  pupil  with  the  literal  operations. 


46  LITERAL  ARITHMETIC. 

Dem. — Let  it  be  required  to  multiply  m,  which  is  either  an  integer  or  a  fra<v 

tion,  by  -. 

1st.  Suppose  a  and  6  are  both  integers.     Multiplying  m  by  a  gives  a  product 

6  times  too  large,  since  we  were  to  multiply  by  only  a  6th  part  of  a ;  hence  we 

am 
divide  the  product,  am,  by  6,  and  have  -r-. 

2d.  When  either  a  or  h,  or  both,  are  fractions.    Let  c  be  the  factor  by  which 

a  a 

numerator  and  denominator  of  r  must  be  multiplied  to  make  -,  a  simple  frac- 
tion (155).  Then  will  ^  be  a  simple  fraction,  i.  e.,  ac  and  he  are  each  integral ; 
and  the  multiplication  is  effected  as  in  Case  1st,  giving  -r—.    This  reduced  by 

dividing  both  terms  by  c  gives  -rr-.    Hence  we  see  that  in  any  case,  to  multiply 

by  a  fraction,  we  have  only  to  multiply  the  multiplicand  by  the  numerator  of 
the  multiplier,  and  divide  this  product  by  the  denominator.  It  is  also  to  be  ob- 
served that  this  reasoning  applies  equally  well  whether  the  multiplicand  is  inte- 
gral or  fractional. 

162,  Cor. — To  multiply  mixed  numhers^  first  reduce  them  to  im" 
proper  fractions. 


Division. 
163.  JProb,  1, — To  divide  a  fraction  hy  an  integer, 
RULE. — Divide  the  numerator  or  multiply  the  denomi- 
nator. 

Dem. — Since  numerator  is  dividend,  and  denominator  divisor,  and  the  value 
of  the  fraction  the  quotient,  this  rule  is  a  direct  consequence  of  {92 f  93). 

16 4 »  Proh,  2, — To  divide  by  a  fractio7i. 

RULE. — Divide  by  the  numerator  and  multiply  the  quo- 
tient BY  THE  DENOMINATOR.  Or,  WHAT  IS  THE  SAME  THING, 
invert  THE  TERMS  OF  THE  DIVISOR  AND  PROCEED  AS  IN  MULTIPLI- 
CATION. 

Dem. — The  correctness  of  the  first  process  appears  from  the  fact  that  division 
is  the  reverse  of  multiplication,  and,  hence,  as  we  multiply  by  the  numerator 
and  divide  by  the  denominator  in  order  to  multiply  by  a  fraction,  to  divide  by 
one  we  must  divide  by  the  numerator  and  multiply  by  the  denominator. 

The  process  of  inverting  the  divisor  and  then  multiplying  by  it  is  seen  to  be 
the  same  as  the  other,  since  this  multiplies  the  dividend  by  the  denominator  of 
the  divisor  and  divides  by  the  numerator. 

Again,  this  process  may  be  demonstrated  thus:  Inverting  the  divisor  shows 


FRACTIONS.  47 

liow  many  times  it  is  contained  in  1.  Then  if  the  given  divisor  is  contained  so 
many  times  in  l,it  will  be  contained  in  5, 5  times  as  many  times  ;  in  f ,  |  as  many 
times  ;  in  cu^ ,  ax^  times  as  many  times  ;  or  in  any  dividend  as  many  times  the 
number  of  times  it  is  contained  in  1,  as  is  expressed  by  that  dividend,  whether 
it  be  integral,  fractional,  or  mixed.     (The  author  prefers  this  demonstration.) 

Sen.  1. — Since  to  multiply  one  fraction  by  another  we  may  multiply  the 
numerators  together  for  the  numerator  and  the  denominators  for  the  denomi- 
nator, and  since  division  is  the  reverse,  we  may  perform  division  by  dividing 
the  numerator  of  the  dividend  by  the  numerator  of  the  divisor,  and  the  de- 
nominator of  the  dividend  by  the  denominator  of  tlie  divisor. 

This  method  will  coincide  with  the  others  when  they  are  worked  by  per- 
forming the  operations  by  division  as  far  as  practicable,  and  this  is  worked 
by  performing  the  multiplications  equivalent  to  the  divisions  when  the  latter 
are  not  practicable. 

16S,  Cor. —  The  reciprocal  of  a  quantify  being  1  divided  by  that 
quantity,  the  reciprocal  of  a  fraction  is  the  fraction  inverted. 

General  Scholium. — In  both  multiplication  and  division  of  fractions,  or  by 
fractions,  all  operations  which  can  be  performed  by  dividing  should  be  so  per- 
formed,  in  order  that  the  result  may  be  in  its  lowest  terms. 


Signs  of  a  Fraction. 

1G6»  In  considering  the  signs  of  a  fraction,  we  have  to  notice 
three  things,  viz.:  the  sign  of  the  numerator,  the  sign  of  the  denomi- 
nator, and  the  sign  before  the  fraction  as  a  whole.  This  latter  sign 
does  not  belong  to  either  the  numerator  or  denominator  separately, 
but  to  the  whole  expression. 

Thus,  in  the  expression  — -—,  in  the  numerator  the  sign  of  4a  is  +, 

and  of  tied  is  — .     In  the  denominator,  the  sign  of  2x  is   +,and  of  A^y^    4-   also, 
'i  he  sign  of  the  fraction  is  — .     These  are  the  signs  of  operation  (50). 


107.  The  essential  character  of  a  fraction,  viS positive  or 
ncyativey  can  be  determined  only  when  the  essential  character  of  all 
th(^  numbers  entering  into  it  is  known.  It  may  then  be  determined 
by  principles  already  given   (78^  97). 


48  literal  arithmetic. 

Examples. 
1.  Reduce  the  following  fractions  to  their  lowest  terms : 


UlOaH^i/^'  bba'^x'     2^^^i^.„  '     1  -x^'   12m^u^  +  2inm^  +  1271^' 

a;~*  -  y-*     a^b  -  1       tg*  —  3rg  —  4      Sa  +  'Saff       3««  +  12a;  4-  9 
.r-i  —  y-i     1  +  aV^'    x^  —  4x  -  5'   4^t__4^t^2'      .r^  +  5x^  +  6  ' 

63:8-3:?:-45     (l+a:)^     n''-''+'//-'c^    2x^y^  +  2         ^x^ +  2x^ -^x 
^x*-\-Vdx^l{fJ\^¥f'     „_p_„,,+,.|'   0:6^6-1' 9^-3 _i;^.c 'J _3o.c  + 48' 

2x*'-x^-  0.r2+  133:-5    2^^^3  4.^^2_  8^^+5r<    a;^- 8:^^ +21a:-18 
lx-^-\\yx-i-\-i^x-b    '        7^3_i2<^2+5^     '     3a;3- 16.^2+ 21a;  ' 
16a;*  —  53a-2  +  45a;  +  6 


8a;*  -  30a;3  +  Sla:^  _  12 


2.  What  factor  will  change    7 to  ^^-r z-~-  ? 

®  a  —  b  a^  —  b^ 

fl*  +  a^x-\-  a*x*  +  fla;3  ■}■  ^-4        n^'  -  x^  .^       x^  +  j^s  +  jg;  4-  1 

a  +  x  a^  -  x^  '  i^'  -  i 

4a;g  -a-  +  4^         6j3*  -  12;x/3  -  0/^3^  _^  127*  ^^  Gf;;^  _  2y3]    ^ 
a;  —  1        *  2)^  —  q^  P  +  Q 

^    ^   ,  1  a;*  4-  14a:*  +  27      J^i*  4.  ,^2  4.  2771?^  —  x  —  y 

3.  Reduce ,     z -. ,     : -, 

1—x  a;2  +  4  m  +  n 

1  _  ^1 0         2         a*  +  4^3,^  4_  Qa^x^  +  4rta;3  +  a;*  a;" 


a  +  1  '     2 -a;'  a^  _,.  3^^2^  4.  3«^2  4.  ^3         '       i_^- 


and 


-^-r: to  inteorral  or  mixed  forms. 


^'  ^^P^'^'^     7^^«'     (^  +  m)-»"     ¥W^^    "^'^    a;-»(«-6)3 
in  the  integral  form. 

X 

5.  Reduce  the  following  mixed  to  fractional  forms :     1  +  :j , 

14a26  +  12a2^,2  -  ^ab  ^       ^         4a;  -  4 

— ,  1  +  2a; -- 

2ab  '  ox 


1  +  7a  +  6«^»  - -^— — ^; ,  1  +  2a;  -      .       , 


^       3a86  4-  2ab^  -  h^        _,       b^  ^  c^  -  a^        ,  ,  1  -  a; 

"-^  +      .2  - b^ — '    1  +  — Wc — '^^^"-^-rr^- 


FBACTIONS.  49 

^    „   ,  3a      4:X     ay     7y        ,    4^^  , 

6.  Reduce    — ,    ;;;^,    -^,    ~,  and  - — -    to  forms  having  a  0.  D. 

7.  Reduce ,    --,  and  -r ^  to  forms  having  a  C.  D. 

a  —  X  a  +  X-        a  —  x         ,       1      ,  -     ^     , 

8.  Reduce ,    —r-^ rr,    ,  and to  equivalent 

X        x(a^  —  x^y    a  i-  X  a  —  x         ^ 

fractions  having  the  L.  C.  D. 

X  X  1'" 

9.  Reduce ,    ^ ~,    and  ; — - — -  to  equivalent  frae- 

tions  having  the  L.  C.  D. 

1  _|_  2;3  (1  —  x)^ 

10.  Reduce  j—- — —  and  ^- ^  to  forms  having  the  L.  C.  D. 

(I  +  x)^  1  —x^  *^ 


11.  Reduce  — ^,  — r,  — -— — -,  and  — r- to  forms  having  the 

L.  C.  D. 

12.  Reduce  — ,  - — — -.,  and  r—^ —  to  forms  having  the  L.  0.  D. 

6x  da;  -t-  4  yrc^  —  16 

13.  Reduce  the  following  complex  fractious  to  simple  ones : 

a  .  fflg  -  ja^  3  ^li^  /^        ~  "^ . 

be     '       ib^y^  +  -2\c^x^ '        3  ^  '       ^    ~7 '        u-^  -ni'^    ' 


c 


b  -^   J      e 

^ ,     .  ^^  a    a  a       ^  a      a;— 7       ,7  —  0;  1  ,        1 

14.  Add  -,  -,  -,  and  -  ;    — ^-  and  -y-;    --^^  and  ^-j--  ; 

a  +  h        ,    a-b-%^                      2  ,  3 

: —    and ;        -^- — and 


2  2  '  x^  +  x^  +x-^\  x^  -  x^  +  x—i' 

a  —  bc  —  a       jJ~c  1  x  +  1  ^x^  +  x  -{-l        1 

1  ,  2flj 

r,  and  -  — ^7^.  4 


a  +  *' a^  -  ^>8 


50  LITERAL  AiarUMETIC. 


lo.  Add  -5 ■ — J,  — — -,  and  ^— — -.  ;  «  —  (  -7-  +  4a*a;^ ) 


2i^  ^  A  J  J 


and  Z>  +  —  +  4a^a;^ 
c 


16.  Add  -,  — -,  and  —  t 7  ;   1 r  and 


a        b~V  b  -\-  I'    I  +  X  +  X*  1  -  X  +  x^  ' 


.T  +  Sa:— 4        ^  X  +  o 
— my Tyy  and  — ■— i-. 


17.  Add  7 r— , jr   and    7 — ; — r-7 — . — ^v  ;      ^ — ^ —    and 


y  -  empy*  .   x  _y_  ^^^^       ^' 
(3my«  —  a;)«  '   y'  a:  4-  y'         a;*  +  a;y' 


18.  Add  -1^,     '      _^-,andi^:^*i^4,i*-£^-±i£^^. 
a  —  0  0  —  cc  —  a  {a  —  0}  {0  —  c)  [c  —  a) 


19.  From    ^  take  -^;   from  i^^    take     i^i^  ; 
a;  —  3  a;  +  3  ao  ab 

from take 


x-7  a; -3* 


20.  From  7a; jj take  x —  ;  from  — 77^  take 

3  2  ab{a  —  by 

ff        ^        c, 
b       a 


21.  From  -n -r-^  —  zttt-, tt  take 


2(a:  +  l)       10  (a:-!)  5  (2a:  +  3)* 


22.  Multiply---   by  --- ;   -— -^  by  -^-^ ;    ^^^ 
4a:2  -  12a:  -  40 


3a-2  -  18a:  +  15 


F1UCTI0N8.  51 

23.  Multiply    ^^—    by  4c;     -  ^x't     by  Jo; 2;      -  ||l'    by 

10^-2       1  +  X  .       ^         o       .       2         «^      I.  <^'""         ^'^"^  t.     2/'^" 

a  ^       "^ 

^'"t 2a:^  +  1  x^ 1     a""  -\-  b'' 2c'* 

24.  Multiply  -J by ;         „_,,     —  by  the 

y^    ^yi^yt^l  yi  _  I  «  ^ 

(^2._.y2)2  ^2-1  gS   -   1 

by  ^^2  _  y'zyz  +  (^2  +  ^2^2  '  (a  +  l)2    by   ^^g  _  ^y 

25.  Multiply  together     -^,     — ;^|^,    and  1  +  ^^. 

26.  Multiply     a:^  —  a;  +  1     by    a;-*  +  a;-i  +  1 ;     1 ~-t    by 

2  + 


b 

a  +  b 
2b 
a—b' 


*'  ax 


27.  Divide -r- by -—- :    — ?— rbym^w*;       ^   „   by  m^n^  :    -r^- 
2     -^  13      m^n^    ^  '    m^n^     ^  '     ^ax 

7^37,T^i  -8-  i.  i  1 

28.  Divide     -^^^-^^-4-^  by   H^'^  V ^^  J    ^^  _  ,^^,    by  1  +  9a» ; 

—    by  a;-"- 

29.  Divide by  1  —  a ;  ^  by —  ;    ( )  by 

1  + «      ^  '  a  +2b    -^  3a  -h  Qb  '    \a      x)     ^ 

(rt  +  x)J^     /     X  1  —  a:\  /     a;      _  1  —  a;\      /c— <^     c^^^sx 

;^  '  vFT^  "^  "IT";  ^  \i  +  a;       ic  y '  Vc+*  c^r^y 

30.  D,v,de    ,«*-«-*  by  ,»  +  -;  \^^^  by  ^^^--p;  a^-h^ 


-cs- 


24,  by  '^-tii ;  f L+_%  +  f )  by  {^^^ ^). 

•'«  +  *  —  c'V  a; +  y       y/    '  \     y  x  +  yj 


52  LITEKAL  ARITHMETIC. 

31.  Divide     {LpiL  by 

1  +  ^ 


{^-y) 


32.  Divide     — ^-^^ -r-ir^ ^^  "l ?• 

«  +  *  «*  +  J* 

33.  Divide     ^  +.i  +  ^  -  3«-'^>-V-'   by  -  +  ^  +  -. 

rt3      ^'      c'  ''  a      0      c 

34.  Free  .,    -^-,        ,    •'    ,    !,-4   ■    '  ^>     ^^^    «^^ 
+  ^~*rt  of  negative  exponents. 

^      ^r,    ,    .    ,,  .  ,     «    6' +  ^/       /5a:3\^     ./m  +  n  , 

35.  What  IS  the  reciprocal  of  (^)  '    '^^      _    >    and 

36.  Is  the  fraction  —  ^-^ j-^  essentially  positive,  or  negative, 

when  a,  in,  x,  and  y  are  each  negative  ? 

Solution. — Since  (—  ay  =  n*,  4«'  is  essentially  positive.     Since  (—  m)(—  a;) 
=  7WJ*,  the  term  Smx,  in  itself,  is  positive,  and   the  numerator  becomes  4a* 

—  (+  dmx),  or  4a*  —  3»w;  (7^)-  Now,  whether  4a*  —  3mx  gives  a  +  or  a  — 
result,  depends  upon  the  numerical  values  of  a,  m,  and  a;.     If  4a*  >  2mx,4a^ 

—  Zmx  is  +  ;  but,  if  4a'  <  3inx,  4a'  —  Snix  is  — .  Again,  since  (—  a;)'  =  —  i»^, 
the  first  term  of  the  denominator,  2x\  is  essentially  negative.  And  since 
(— y)^  =  y*,  the  term  4y-  is  essentially  positive  and  the  denominator  becomes 

—  2x^  4-  (  +  4y'),  or  —  2j;*  +  4y'.  Whether  this  is  +  or  — ,  depends  upon  the 
relative  values  of  x  and  y.    If  we  suppose  4a*  >  3mx  the  numerator  becomes  + , 

and  if  2^;"'  be  greater  than  4y*  the  denominator  becomes  — ,  and  we  have  — — , 
which  gives  a  positive  result. 

37.  What  is  the  essential  sign  of r-^ j-,  when  «=  —1,  b=2, 

°  abxy  —  4 

x=  —3,  and  y  =  —  4 ? 

1 

3^^^  3  7J"3'|/ 

38.  What  is  the  essential  sign  of  — __  . , — -y  when  a  =  —  3, 
i  =  —B,  771  =  —1, and  y  =  I? 


FRACTIONS.  53 

2a^x^ daJb 

39.  What  is  the  essential  sign  of j j,  when  a=  —32, 

h  =  —  2, 771  =  —  S,  and  x  =  —2? 

40.  Simplify 


X  + 

1    \  ,  1 

y{xyz  +  cc  • 

+  ^)' 

1 

y 

1 

1 

X      a  —  y 

+  (^ 

•  X 

J' 

a  — 

-  xY       (a 

-  ?/)' 

1 

1 

(a 

3 

»;)» 

(«-y)M« 

-X) 

1 

1         1 

3- 

a 

and 

be      ca      (lb 

a  + 

b 

-b*) 

1. 

a  — 

-1 

64  LITERAL  AlilTHMETIC. 


CHAPTER  IV. 

POWERS  AND  BOOTS. 


SECT/ ON  L 
INVOLUTION. 


Definitions. 

l^SS.  A  ^ower  is  a  product  arising  from  multiplying  a  number 
by  itself.  The  Degree  of  the  power  is  indicated  by  the  number 
of  factors  taken. 

ScH. — It  will  be  seen  that  a  power  is  a  species  of  composite  number  in 
which  the  component  factors  are  equal. 

i(>.9.  A  Root  is  one  of  the  equal  factors  into  which  a  number  is 
conceived  to  be  resolved.  The  I^egree  of  the  root  is  indicated 
by  the  number  of  required  factors. 

170.  An  Exponent  or  Index  is  a  number  written  a  little 
to  the  right  and  above  another  number,  and 

1st.  If  a  Positive  luteffpr.  it  indicates  a  Power  of  the  number; 

2d.  If  a  Positive  Fraction,  the  numerator  indicates  a  Power,  and 
the  denominator  a  Root  of  the  number ; 

3d.  If  a  Negative  Integer  or  Fraction,  it  indicates  the  Reciprocal 
of  what  it  would  signify  if  positive. 

ScH, — It  is  obviously  incorrect  to  read  4%  "the  f  power  of  4."  There  is 
no  such  thing  as  a  2-fifths  power,  as  w^ill  be  seen  by  considering  the  defini- 

m  ni 

2.  -  m  —  - 

tion  of  a  power.     Read  4%  "4  exponent  | ;  "  also  a" ,  "«  exponent  ^ ; "  a   " , 

"«  exponent  —  ^."     These  are  abbreviated  forms  for,  "«  with  an  exponent 

—  ^, "  etc.     In  this  way  any  exponent,  however  comphcated,  is  read  witliout 
difficuay. 


POWERS  AND   ROOTS.  55 

17 !•  A  Radical  dumber  is  an  indicated  root  of  a  number. 
If  the  root  can  be  extracted  exactly,  the  quantity  is  cnlled  Rational ; 
if  the  root  cannot  be  extracted  exactly,  the  expression  is  called  Irra- 
tional, or  Surd. 

172,  A  Root  is  indicated  either  by  the  denominator  of  a  frac- 
tional exponent,  or  by  the  JRadlcal  Sigti,  V.  This  sign  used 
alone  signifies  square  root.  Any  other  root  is  indicated  by  writing 
its  index  in  the  opening  of  the  v  part  of  the  sign. 

173,  An  Imaginary  Quantity  is  an  indicated  even  root 
of  a  negative  quantity,  and  is  so  called  because  no  number,  in  the 
ordinary  sense,  can  be  found,  which,  taken  an  even  number  of  times 
as  a  factor,  produces  a  negative  quantity. 

Thus  V  —  4  is  imaginary,  because  we  cannot  find  any  factor,  in  the  ordinary 
sense,  which  multiplied  by  itself  once  produces  —  4.  Neither  +  2  nor  —  2  pro- 
duces —  4  when  squared.  For  a  like  reason  V  —  da"^,  V  —  5x,ot  >y/—  14^xy* 
are  imaginaries. 

174,  All  quantities  not  imaginary  are  called  Real 

17 5,  Similar  Radicals  are  like  roots  of  like  quantities. 

Thus  A>/5a,  ^y/5a,  and  (a*  —  x^)V5a  are  similar  radicals. 

176,  To  nationalize  an  expression  is  to  free  it  from  radicals. 

177,  To  affect  a  number  with  an  Exponent  is  to  per- 
form upon  it  the  operations  indicated  by  that  exponent. 

178,  Involution  is  the  process  of  raising  numbers  to  required 
powers. 

179,  Evolution  is  the  process  of  extracting  roots  of  numbers. 

180,  Calculus  of  Radicals  treats  of  the  processes  of  re- 
ducing, adding,  subtracting,  or  performing  any  of  the  common 
arithmetical  operations  upon  radical  quantities. 


Involution. 

181,  I^rob,  1, — To  raise  a  number  to  any  required  power, 
RULE. — Multiply  the  number  by  itself  as  many  times,  less 

ONE,  AS  THERE  ARE  UNITS  IN  THE  DEGREE  OF  THE  POWER. 

182.  Cor. — Since  any  nmnher  of  positive  factors  gives  a  positive 
product,  all  powers  of  positive  monomials  are  positive.     Again, 


66  LITERAL  AKITHMETIC. 

since  an  even  number  of  negative  factors  gives  a  positive  product^ 
and  an  odd  number  gives  a  negative  product,  it  follows  that  even 
powers  of  negative  numbers  are  positive^  and  odd  powers  negative. 

183,  JProb.  2, —  To  affect  a  monomial  with  any  exponent. 
RULE. — Perform    upon   the   coefficient    the    operations 

INDICATED  BY  THE  EXPONENT,  AND  MULTIPLY  THE  EXPONENTS  OF 
THE   LETTERS   BY  THE  GIVEN  EXPONENT. 

Dem. — iBt.  When  the  exponent  by  which  the  monomial  is  to  be  affected  is  a  positive 

n 

integer.    Let  it  be  required  to  affect  ia^b"  x-  •  with  the  exponent  p;  or  in  other 
words  raise  it  to  the  pth  power,  p  being  a  positive  integer.    The  pi}\  power  of 
*  "  "  " 

^"'b'^  x-^  is  ia^b"^ x-"  x  4a"'b'  x-*  x  ^"^b"^  x-" to  p  factors.      But  as 

the  order  of  the  arrangement  of  the  factors  does  not  affect  the  product  (77), 
this  product  may  be  considered  as,  p  factors  each  4,  into  p  factors  each  a**,  into  p 

factors  each  b%  into  p  factors  each  x-*.    Now  p  factors  each  4  give  4''  by  definition. 

p  factors  each  a""  are  expressed  a^'",  since  a*  is  m  factors  each  a,  and  p  factors  con 

taining  m  factors  each,  make  in  the  whole  pm  factors,  or  a^'"».     Again,  p  factors 

£?  *  1 

each  &  ■■  are  expressed  b  ^ ,  since  6 '  is  n  factors  each  b " ,  and  p  factors,  containing  n 

-         —  1  11 

factors  each,  are  pn  factors  each  6 ' ,  or  6  »■ .   And  since  xr*— — ,  p  factors,  or  —  x  — 

Xf  iff     X* 

X  _  .  .  .  to  p  factors  make  — ,  as  fractions  are  multiplied  by  multiplying 
numerators  together  for  a  new  numerator  and  denominators  for  a  new  denomi- 
nator, and  J?'  X  ar*  X  «»-  -  -  to  »  factors  are  xf"".    But  —  =  x-p".    Hence  collect- 

n  P* 

ing  the  factors  we  find  that  (4a"'yx~'')P  =  4''a^"*&'^  x-"'.    q.  E.  D. 

2d.   When  the  exponent  is  a  positive  fraction.    Let  it  be  required  to  affect 

4a'^b'^  x-',  with  the  exponent  — .  This  means  that  Aa'^b'^x-*  is  to  be  resolved 
into  q  equal  factors  and  p  of  them  taken.     Now,  if  we  separate  each  of  the  fac- 

n 

tors  of  4a"*6 ^  x-'  into  q  equal  factors, and  then  take  p  of  each  of  these,  we  shall 

have  done  what  is  signified  bv  the  exponent  ~. 

<? 
1 . 

By  definition,  4  '  represents  one  of  the  q  equal  factors  of  4. 

To  obtain  one  of  the  q  equal  factors  of  a"s  we  take  one  of  the  q  equal  factors 


POWERS   AND   ROOTS.  57 

of  a  from  each  of  the  m  factors  represented.     But  one  of  the  q  equal  factors  of 
1  »' 

a  is  represented  hy  a'' ,  and  m  of  these  is  a'  by  definition. 

n  n 

To  separate  }/  into  q  equal  factors,  we  notice  that  l{  is  ?«,  of  the  r  equal  fac- 
tors of  h.  Now,  if  we  resolve  each  of  these  r  factors  into  q  equal  factors,  h  is 
resolved  into  rq  equal  factors ;  doing  the  same  with  each  of  the  n  factors  repre- 
sented, and  taking  one  from  each  set,  we  have  h  resolved  into  rq  equal  factors 

and  n  of  them  taken  ;  that  is  6*^*  is  one  of  the  q  equal  factors  oth^ . 

1 
To  resolve   «-»=  —  into  q  equal   factors,  we   consider  that  a   fraction  is 

resolved  by  resolving  its  numerator  and  denominator  separately.     But  one  of  the 

q  equal  factors  of  1  is  1 ;  and  one  of  the  q  equal  factors  of  -x^  is  x^  as  seen  in  the  re- 

1         1         -- 
solution  of  a"*.    Hence  one  of  the  q  equal  factors  of  x-i  or  jL  is  ~7  •=  -j    "^ . 

n 

Collecting  these  factors  we  find  that  one  of  the  q  equal  factors  of  4n!'»&'.c-''  is 

1    m     n    _  ♦ 

4'3'^'j'5<}r^   «.    And  finally  j?  of  these  being  obtained  according  to  Case  1st,  gives 

p     pm  Tt*       p»  n 

41  ^7  ^TT^   « ^  ag  the  expression  for  4a"'6'^  a;-*  affected  with  the  exponent  ?;  which 

g 

result  agrees  with  the  enunciation  of  the  rule. 

3d.   When,  tJie  exponent  is  negative  and  either  integral  or  fractional.      Let 

n 

it  be  required  to  affect  4«'"6'"i;-«  with  the  exponent  —t.     This  by  the  definition 
of  negative  exponents,  signifies  that  we  are  to  take  the  reciprocal  of  what  the 

J* 
expression  would  be  if  t  were  positive.    But  Aa"'h^x-*  affected  with  the  exponent 

t  (positive)  is  4'a'"'6'' «-'■"'   by  the  preceding  cases,  whether  t  is  integral  or  frac- 
tional.   The  reciprocal  of  this  is .     But   since  these   factors  can  be 

A'a'"'hTx-*» 
transferred  to  the  numerator  by  changing  the  signs  of  their  exponents,  we  have 

'*  * 

4-'a-'"'6   "^'j^",  as  the  result  of  affecting  4a"'b''x-'  with  the  exponent  —t,  which 

result  agrees  with  the  enunciation  of  the  rule. 


184,  I^rob*  3» — To  expand  a  binomial  affected  with  any  expo- 
ne?it. 

liULB.—TuiS    RULE    IS    BEST  STATED    IN  A    FORMULA.      ThUS, 
LET     rt,    bf   AND   m     BE     ANY     NUMBERS     WHATEVER,     POSITIVE    OK 


68  LITERAL     ARITHMETIC. 

NEGATIVE,   INTEGRAL  OR   FRACTIONAL,   THEN   WILL    (flf-f  *)"*  REPRE- 
SENT  ANY   BINOMIAL,   AFFECTED   WITH   ANY   EXPONENT,  AND 

(a  +  by  =  a*"  +  7na'^-^  +  ^  ^f^  ~  ^^  rt'»-2^>2 

1     .    /i 

m  {m  -  1)  {m  -  2) 
+       1     •     2    •     3"'''      ^ 

m(m-l)(m-2)(m-3)    .._ 
+         1     •    2    •    3     •    4        ^      ^ 

m  (m  -  1)  (m  -  2)  (m  -  3)  (;//  -  4) 
+  1     •    2     •    3     •    4    •    5  «     ^,  +  etc. 

This  is  the  celebrated  Binomial  Formula,  or  Theorem.  Its  demonstra- 
tion will  be  found  in  the  subsequent  part  of  the  work.  At  this  stage  of  his 
progress  the  student  should  learn  the  formula  and  become  expert  in  applying  it. 

18o,  CoR.  1. —  T/te  exjXf9isio9i  of  a  binomial  terminates  only  when 
the  ex'ponent  is  a  positive  integer^  since  only  when  m  is  a  positive 
integer  will  a  factor  of  the  form  in(m  —  1)  (m  —  2)  (m  —  3),  etc., 
become  0,  as  is  evident  by  inspection. 

186,  Cor.  2. —  lV7ie7i  m  is  a  positive  integer,  that  is  when  a  bino- 
mial is  raised  to  any  power,  there  is  one  more  term  in  the  develop- 
ment than  xinits  in  the  exponent. 

Since  the  first   coefficient  is   1;  the  2d,  m\    the   3d,       ^     ~      ;    the   4th, 

mim,  —!)(»» —  2)      ^,      ^,,     w(w  —  l)(m  —  2)  (w  —  3)      .  ^.       ^.    ^  ^- 

^      — ^-^— ■' ;    the  5th,  — '— -     ,  /     . ;  etc.,  we   notice   that  the 

3         *         O  i  '  O       '         '*■ 

last  factor  is  m  —  (the  number  of  the  term  —  2) ;  and  the  number  of  the  term, 
therefore,  which  has  m  —  m  as  a  factor  is  the  (w  +  2)th  term.  But  this  is  0. 
Hence  the  {m  +  l)th  term  is  the  last. 

187,  Cor.  3. —  When  m  is  a2)ositive  integer,  the  coefficients  equally 
distant  from  the  extremes  are  equal. 

Thus  {a  +  ft)*  =  (6  +  a)"*;  the  former  of  which  gives  a"'  +  ma'"-^b  + 
ffl(m  -  1)^^_2^8  _j_^   g^p^  ^^^  ^j^g  j^^^^,j.  j«  ^  mb'"-'a  +  ^^^~  ^V"-V  +,  etc. 

Whence  it  appears  that  the  first  half  of  the  terms  and  the  last  half  are  exactly- 
symmetrical. 

188,  Cor.  4:.  — The  sum  of  the  exponents  i7i  each  term  is  the  same 
as  the  expo7ient  of  the  power. 

Sen. — The  last  two  corollaries- apply  to  the  form  {x  +  yy\  and  not  to  such 
forms  as  (2a'  —  35-)'",  after  the  latter  is  fully  expanded. 


POWERS   AND   ROOTS.  ^ 

1S9,  Cor.  5. — A  convenient  rule /or  writing  out  the poioers  of 
binomials  may  be  thus  stated: 

\st.  There  is  one  more  term  in  the  development  than  there  are 
units  in  the  exponent  of  the  power. 

2d  The  FIRST  contains  only  the  first  letter  of  the  binomial^  and  the 
last  term  only  the  second^  while  all  the  other  terms  contaiji  both  the 
letters. 

3d.  T/ie  exponent  of  the  first  letter  of  the  binomial  in  the  first  term 
of  the  development  is  the  same  as  the  exponent  of  the  required  power 
and  DIMINISHES  by  unity  to  the  right,  xchile  the  exponent  of  the 
second  letter  begins  at  unity  in  the  second  term  of  the  ex^^ansio?!  and 
INCREASES  by  unity  to  the  right,  becouiing,  in  the  last  term,  the  same 
as  the  exponent  of  the  power. 

4tth.  The  coefficient  of  the  first  term  of  the  expansio7i  is  unity  ;  of 
the  second,  the  exponent  of  the  required  poyner  ;  and  that  of  any  other 
term  may  be  found  by  mxdtiplying  the  coefficient  of  the  preceding 
term  by  the  exponent  of  the  first  letter  in  that  term,  and  dividing  the 
product  by  the  exponent  of  the  second  letter  +  1. 

190,  Cor.  6. — If  the  sign  betweoi  the  terms  of  the  binomial  is 
jninus,  as  (a  —  b)°*,  the  odd  terms  of  the  expansion  are  +  and  the 
eve7i  ones  — .  This  arises  from  the  fact  that  the  odd  terms  involve 
even  powers  of  the  second  or  negative  term  of  the  bhiomial,  and  the 
even  terms  invclve  the  odd  powers  of  the  same. 


Examples. 

1.  What  is  the  square  of  3^3  ?    Of  -2fAc  ?   Of  ^x~^^  Of  -^a^x  ? 
Oi^^/x'i     OfiV2?     Of-^? 

2.  What  is  the  square  of  1  -  x  +  x'-  ?     Of  2a  -  3x^  ? 

2  3      ,  3 

3.  Expand  the   following:  (3-1x-x^)  ,  {3x^ -  1)  ,  {x-y  +  z)  , 
{l~x^),  {x^-yh'- 

4.  Aifeofc  3rAr«  with  the  exponent  4;  ^a^x^  with  tlie  exponent  2; 

a'^x  with  the  exponent  ~m,vi'\t\\  the  exponent  J,  f ;  bx^y  with  the 
exponent  |,  ^,  —  3. 


00  LITERAL   ARITHMETIC. 

5.  Perform  the  following  operations  and  explain  each  as  a  process 

3  8  ^ 

of    factoring,    according   to   (De^h.ISS):     (12ba^x^)^,      (Ua^xy, 

-4       3  -4  4    1-4-1^  •*_'_" 

6.  Expand  the  following  by  the  Binomial  Formula:    (x -\- yY, 
{x-yY,  (3«2-.t)3,  {x  +  y)-^,  (^-y)"*,  (5  +  :c2)*    (a:^-^^)'^ 

77(free  results. 

\/«*  —  a^e^=  rtVl  —  «-  =  ^(1  —U^  —  ,r--;g*  — »-  .    ^e«— etc.) 

\         »  2*4        2 '4 '6  ' 

(1  —  .T«)'^  =  1  +  ^a:2  +lx*'  +  A^«  +  i%a:8  ^^ etc. 


(fl?«  +  hx^y  =a  + — —   4-  z^—.  -,  etc. 

7.  Write  out  by  Cou's.  5   and  6,  the  expansions  of  the  follow- 
ing: (a  +  l)y,{a-by,{a^'-b'^)\  (x^ -y^Y,(a^  ^y^)\  {x'^ -y^y. 


SECTION  II. 

EVOLUTION. 

191,  I^rob,  1. — To  extract  any  root  of  a  perfect  power  of  that 
degree. 

RULE. — Kesolve  the  number  into  its  prime  factors,  and 

SEPARATE  THESE  INTO  AS  MANY  EQUAL  GROUPS  AS  THERE  ARE 
UNITS  IN  THE  DEGREE  OF  THE  ROOT  REQUIRED;  THE  PRODUCT  OF 
ONE   OF   THESE   GROUPS    IS   THE   ROOT   SOUGHT. 

192,  Sen. — The  sign  of  an  even  root  of  a  positive  number  is  ambiguous 
(that  is  4-  or  — ),  since  an  even  number  of  factors  gives  the  same  product 
whether  they  are  positive  or  negative  (79 f  80).  The  sign  of  an  odd  root  is 
the  same  as  that  of  the  number  itself,  since  an  odd  number  of  j^ositive  factors 
gives  a  positive  product  and  an  odd  number  of  negative  factors  gives  a 
negative  product  (80,  81). 

193,  Cor.  1. —  The  roots  of  monomials  can  be  extracted  by 
extracting  the  required  root  of  the  coefficient  and  dividing  the  expo- 
nent of  each  letter  by  the  index  of  tfie  root^  since  to  extract  the  square 


POWERS   AND   ROOTS.  ^1 

root  is  to  affect  a  number  with  the  exponent  \^  the  cube  root  ^,  the  nth 
root  i,  etc.  (183). 

104,  Cor.  'Z. —  71ie  root  of  the  product  of  several  numbers  is  the 
same  as  the  product  of  the  roots. 

Thus,  "Vobcx  =  "Va  ■  'Vb    Vc  •  V.T,  since  to  extract  the  mih  root  of  obex 
Ave  have  but  to  divide    the    exponent    of    each   letter    by    m,  which  gives, 

ILil  _        _       _        _ 

a"'b^c^x"^,  or  Va  ■  Vb  ■  Vc  ■  Wx. 

IQS,  Cor.  3. —  The  root  of  the  quotient  of  two  numbers  is  the  same 
as  the  quotient  of  the  roots. 


Thus,   4/'  —  is  the  same  as    _^,  since  to  extract  the  rih.  root  of  —  we  liave 


but  to   extract  the  7*th  root  of    numerator  and  denominator,  which  operation 

is  performed  by  dividing  their  exponents  by  r.    Hence    a/'HI  —  — r  ::= !^. 

r    n  —      W^ 


nr        >^^ 


Examples. 

1.  Extract  the  square  root  of  each  of  the  following  numbers  by 
resolving  them  into  their  factors,  i.  e.  by  (191) :  222784 ;  2131G ; 
and  5499025. 

2.  Extract  the  square  root  of  each  of  the  following,  as  above : 
81«*a:-'?/V%  a^c^-\-2a^bc^  +a^b^c^,  m^—2m*x  +  7n*x^. 


3.  Extract  as  above :   V^oa^b'*,  V  G4rr«ic%  y49a;y%  \^lUa^m^f 


'49rt 


^,  yHJn  ^7/8,  ^125m».Ti«,  yiTZSx^y^,  ^/ -'d'Za^^y-K 


y  36m2/i 

4.  Solve  exercises  2  and  3  also  by  (193)- 

5.  Show   as  in   (194)   that    >v^8  x  27  =  V^xv^27;    also  that 


6/  \_  .  5/    1_ 

y  a-'^b*  =  ya""  x  -(/  i". 


6.  Is  V«±^=  V«±V^?     Is  i /^=-i?    Is   Vab=VaVb? 

y  ^  Vb 

Why  does  the  reasoning  in  the  cases  which  are  true  not  apply  to  the 
others?    State  the  true  propositions ;  also  the  false  assumption. 

ScH. — The  extraction  of  roots  by  resolving  numbers  into  their  factors 
according  to  ihi»  rule,  is  limited  in  its  application  for  several  reasons.     In 


62  LITERAL   ARITHMETIC. 

the  case  of  decimal  numbers  we  can  always  find  the  prime  factors  by  trial, 
and  hence  if  the  number  is  an  exact  power,  can  get  its  root.  But  in  case 
the  number  is  not  an  exact  power  of  the  degree  required,  we  have  no  method 
of  approximating  to  its  exact  root  by  this  rule,  as  we  have  by  the  common 
method  already  learned  in  arithmetic.  In  case  of  literal  numbers  the  diffi- 
culty of  detecting  the  polynomial  factors  of  a  polynomial  is  usually  insuper- 
able. Hence  we  seek  general  rules  which  will  not  be  subject  to  these 
objections. 


196 •  JProb,  2, — To  extract  roots  whose  indices  are  composed  of 
the  factors  2  and  3. 

Solution. — To  extract  the  4th  root,  extract  the  square  root  of  the  square  root. 
Since  the  4th  root  is  one  of  the  4  equal  factors  into  which  a  number  is  conceived 
to  be  resolved,  if  we  first  resolve  a  number  into  2  equal  factors  (that  is,  extract 
the  square  root)  and  then  resolve  one  of  these  factors  into  2  equal  factors  (that 
is,  extract  its  square  root)  one  of  the  last  factors  is  one  of  the  4  equal  factors 
which  compose  the  original  number,  and  hence  the  4th  root.  In  like  manner 
the  Gth  root  is  the  cube  root  of  the  square  root,  etc. 


107.  JProb.  3, —  To  extract  the  mth  {any)  root  of  a  number. 

Solution. — Instead  of  giving  in  detail  the  demonstrations  of  the  processes  for 
the  extraction  of  roots,  we  assume  that  the  student  is  familiar  with  the  subject 
as  presented  in  common  arithmetic,*  and  propose  here  to  show  him  how  to  see 
a  rule  for  the  extraction  of  any  root  of  a  decimal  number,  and  of  a  polynomial, 
in  the  expansion  of  a  binomial.     Thus 

For  the 

Square  root        (a  +  6)*  =  a'  +  (2a  +  6)6  gives  the  rule ; 

Cube       "           (a  +  &)^'=«='  +  (3a«+3a6  +  62)6  "        "      " 

Fourth    "          (a  +  &)*=a*  +  (4a«  +  6a«6  +  4<i6«+6')6  "        "      " 

Fifth       "          (a  +  &)'=«'+ (5a*  +  lOa'ft  +  10a*&«  +5ofZ>'^  +  6*)6  "        "      " 
etc.,                etc.,                etc. 

In  all  cases  a  represents  the  part  of  the  root  already  found,  and  h  the  next 
figure  or  term  of  the  root ;  observing  that  in  decimal  numbers,  a  is  tens  with 
reference  to  b. 

The  method  of  pointing  off  decimal  numbers  into  periods,  and  the  reason, 
are  shown  for  the  square  and  cube  root  in  common  arithmetic ;  and  the  same 
reasoning  extends  to  other  roots. 

A  polynomial  must  be  arranged  as  for  division,  since  this  is  the  form  which, 
a  power  takes  when  the  root  is  similarly  arranged. 

The  solution  of  a  few  examples  will  familiarize  the  student  with  this  method. 
♦  The  wh(rie  subject  is  fully  presented  in  the  Complete  School  Aloebba. 


powers  and  roots. 

Examples. 
1.  Extract  the  square  root  of  7284601. 

SOLUTION. 

The  fonnula  is  (a  -f  &)*  =  n^  +  {2a  +  b) b. 
At  first  a^  =  the  greatest  square  in  7.     .".  a  =  2. 


I  7284601 12699 

4 


2f(,  =  2(20)  =  the  Trial  Divisor 40  j 

.•.  328  -f-  40  =  8  is  the  jJi'obaJde*  second  root  figure, 
(2ti  +  6)  =  40  +  8  is  the  True  Divisor  if   8  is  the  second  root 
figure.     But  48  x  8  =  384.     .-.  8  is  too  large.     We  will  try 

6  as  the  second  root  figure 6 

Whence  (2a  +  6)  =  the  True  Divisor 46 


328 


276 


J!^otc,  2a  =  2  (260)  =  the  2Hal  Divisor 520  j  5246 

.'.  5246  -h  520  =  the  probable  next  root  figure 9  I 

{2a  +  6)  -  520  +  9  =  the  True  Divisor 529  I  4761 

Again,  2a  =  2  (2690)  =  the  Trial  Divisor 5380  I  48501 

.-.  48501  ■+■  5380  =  the  probable  next  root  figure 9 

{2a  +  b)  =  5380  +  9  =  the  True  Divisor 5389  I  48501 


2.  Extract  the  cube  root  of  99252847. 

SOLUTION. 

The  formula  is  {a  +  b)^  =  a^  +  (3a«  +  Sab  +  b*)b. 
At  first  a'  =  the  greatest  cube  in  99.     .'.  a  —  4. 


3a*  =  3  (40)*  =  the  THal  Divisor 4800 

.-.  35252  -4-  4800  =  7,  Wxa  probable  next  root  figure. 
(3a*  +  3rt6  +  &*)  =  4800  +  840  +  49  =  5689,  the  True  Divisor 
if  7  is  the  next  root  figure.     But,  as  this  does  not  go  7 
times  in  35252,  7  is  too  large  ;  and  we  try  6. 
Noic,  the  corrections  to  be  added  to  the  trial  divisor  to  make 

the  true  divisor,  are  3a6  =  3  (40)  6  -     720 

and        b'  =  (6)^       -      86 


992528471463 
64 

35252 


Hence  the  true  divisor  is 5556  j  33336 


New  THal  Divisor,  3a*  -=  3  (460)*  = G348(X)  i  1916847 

j3a6:=3(460)3  = 4140  ! 

Corrections:^   &^  ==  (3)*  = _     9! 

True  Divisor 638919  I  1916847 


*  The  new  root  figure  cannot  be  larger  than  this  quotient.    It  is  often  not  eo  large,  and  the 
probability  of  its  being  considerably  less  increases  with  the  degree  of  the  root  wc  arc  extracting,. 


^  LITERAL   ARITHMETIC. 

3.  Extract  the  5th  root  of  30036242722357. 


SOLUTION. 


Formula :    {a  +  by  =  a'  +  5a*b  +  10«='6«  +  10^*6='  +  Bab*  +  ft» 

=  a^  +  [5a*  +  lOa'^b  +  lOa'b'  +  Bab''  +  b*]b. 


At  first  a*  =  the  greatest  5th  power  in 


Trial  Divisor :  5a*  =  5  (50)*  = 

1st.  lOa'ft  =  10(50)'  X  1  =  . 
2d.  10rt«6*  =  10(50/  X  1^  = 
3d.  5a*^  =5(50)  x  !=»  =  .... 
4th.  b*  =  V  = 


Corrections :  - 
Tnie  Divisor 


31250000 

1250000 

25000 

250 

1 

¥2525251 


36936242722357|517 

3125 

56862427 


32525251 


Ti-ial  Divisor :  5a*  -  5  (510)*  = 338260050000 

1st.  10a '6  =  10  (510) '  X  7  =  . . .  9285570000 
2d.  10a«6«  =  10  (510) «  x  7'  =  .    127449000 

3d.  Bab'  =  5  (510)  x  7»  = 874650 

4th.  6*  =  7*  = 2401 

Tnie  Divisor : 847673946051 


Corrections :  - 


2433717622357 


2433717622357 


4.  What  is  the  7th  root  of  1231171548132409344  ? 


SOLUTION. 

Formula:  (a  +  6)^  =  a'+7a«6  +  21a»6*  +  9Ba*b' +  SBa^'b*  +  2U^b'' -h  7ah^  +  b' 
=  a'  +  I7a«  +  21a»6  4-  dBa*b'  +  SBa'b^  +  21a*6*  +  7a*'  +  b^]b. 


1231171548132409344|384 

2187 


Trial  Divisor :    7a6  =  7  (30)^  = 5103000000 

..  fist.  21a»6  =  21(30)»  X  8= 4082400000 


2d.  35a*6*  =  35(30)*  x  8*  =  . 
3d.  SSrrVy^' =  35(30)-'  x  8^  =  . 
4th.  2la^b*  =  21  (30)*  x  8*  = , 

5th.  7aA«=:7(30)  x  8^  = 

L6th.  66  =  8'^  = 


,  1814400000 

,  483840000 

77414400 

0881280 

262144 

11568197824 


101247154813 


92545582592 


Trial  Divisor :  7a«*  =  7  (380)*^  = . . 

.21076564688000000 

87015722212409344 

.. 

f  1st.  21a'b  =  21  (380)'  x  4  = . . 

.  Ga557541 1200000 

02 

2d.  Soa^y-  =  35  (380)*  x  4«  = 

.   11676761600000 

*■? 

3d.  35a^'6-^  =  35(380)='  x  4='  = 

122913280000 

g 

4th.  21a'6*  =  21  (380)«  x  4*  = 

776294400 

o 

5th.  7a*'  =  7  (380)  x  4'  = . . . 

2723840 

6th.  56  ^  4<5  ~ 

4096 
2i7o3930o53102336 

87015722212409344 

POWERS  AND   ROOTS.  6© 

5.  Extract  the  square  root  of  each  of  the  following  numbers :  7225, 
9801,  553536,  5764801,  345642,  2,  .5,  3,  50,  1.25,  1.6. 

6.  Extract  the  cube  root  of  each  of  the  following  numbers :  74088, 
122097755681,  2936.493568,  61234,  12.5,  .64,  .08,  2,  5. 

7.  Extract  the  4th  root  of  52764813.     (See  196.) 

8.  Extract  the  6th  root  of  2985984.     (See  196.) 

9.  Extract  the  8th  root  of  1679616.     (See  196.) 

10.  Extract  the  5th  root  of  5.     \/5"=  1.37974  -. 

11.  Extract  the  7th  root  of  2.     -^^2  =  1.104  +  . 

12.  Extract   the   square  root  of  49a:8|/«  —  30a;3y  +  16y*— 24a;y» 
■f  25a:*. 

SOLUTION. 

Formula  :  {a  +  hy  =a'  +  {2a  +  h)b. 
:.  a=^x*  25«^ 


2a=  Trial  Div.  =     10;c''    -'dOx'y  +  AQx'y^ 
b=  -30^^^-f-lO.c'  =  -3.ry 
.-.  True  Div.:=10.c-  —  Sxy 


-30x'*p+  9a;'.?/ 


i^it 


2a=  Trial  Div.  =          10.c* — 6.r^ 

4Qx'y'-24xy'  +  lQy* 

.-.  *=40j;V«-4-10.c*=             4y2 

and  True  Div.=iac«-6a^  +  4y' 

40;rV-24i:y^  +  16y* 

CONDENSED    SOLUTION. 

25a;*  -30a; 'y  +  49a;'y«  -24^2/='  +  16y*  |5a;^-3a^  +  4y* 
25x*  " 


10x^-3x1/ 


■30a;V  +  49a;V' 
-30.2; '.v+   9a;«.v« 


lOx'— 6a^  +  4^/• 


40a;V'-34a^'+16y^ 
40a;^y«-24a;y=»  +  16y* 


66 


LITERAL   AlUTHMETIC. 


o 

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E2  (S 


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f 

POWERS  AND  ROOTS.  67 

15.  Extract  the  cube  root  of  each  of  the  following:  a^ -- Sb^ 
+  VZab^  —  6a^b,  5^:3  —  1  -  Sx^  +  x^  —  ^x,  66x*  +  1  —  63a;3—  9a; 
+  Sx^  -  36a:«  +  33a;2,  mc^x^  +  ^^cx^  -  %lc^  +  lO^o^x  -  '^{)c^x^ 
+  8a;«  —  806'3a;3,  204:C^x^  —  lUc^x  +  8a;6  -  366-^«-  171c^x^ +  64.a^ 

+  102c^x^,   27a;  -  8a;*  -  36  +  362;^  +  12a;-^  -  54a;*  +  9a;"*  +  27a;* 

4-a;-"— 6a;'*. 

16.  What  is  the  4th  root  of  16«*  -  96^3:2;  +  216«2a;2  -  216aa;3 
4-  81a;*  ? 

17.  What  is  the  6th  root  of  729  -  2916a;2  +  4860a;*  -  4320a;« 
+  2160a;8- 576a;i«+  64a;i2? 

[Note.— Solve  the  16th  and  17th  both  by  {197)  and  {196)1 

18.  Find  the  fifth  root  of  32a;5-  80a;4  +  80a;3-  40a;2  +  10a;  -  1 ; 
also  of  a;-"  +  15a;-"-5a;-'*  +  90a;-"-60a;-'»  +  280a;-'-270a;-«  +  495a;-* 
-  550a;-'  +  513  -  465a;2  +  275a;*  -  90a;«  +  15a;8  -xAK 

19.  Find  the  6th  root  of  ««-  6a^b  +  15a*^2-  20^3^3  4.  15^2^4 
-6a*6+J«by  (196). 


SECTION  IIL 
CALCULUS  OF  RADICALS. 


Reduction. 
19S,  Pvob,  !• — To  simplify  a  radical  by  removing  a  factor, 
RULE. — Resolve  the  number  under  the  radical  sign  into 

TWO  FACTORS,  ONE  OF  WHICH  SHALL  BE  A  PERFECT  POWER  OF  THE 
DEGREE  OF  THE  RADICAL.  EXTRACT  THE  REQUIRED  ROOT  OF  THIS 
FACTOR  AND  PLACE  IT  BEFORE  THE  RADICAL  SIGN  AS  A  COEFFICIENT 
TO  THE   OTHER  FACTOR   UNDER  THE   SIGN. 

Dem, — This  process  is  simply  an  application  of  CoR.,  Art.  194:, 

199,  Cor. —  The  denominator  of  a  surd  fraction  can  ahcays  be 
removed  from  under  a  radical  sign  by  multiplying  both  terms  of  tha 
fraction  by  some  factor  which  icill  make  the  denominator  a  perfect, 
power  of  the  degree  required. 


BS  LITERAL  ARITHMETIC. 

ScH. — A  surd  fraction  is  conceived  to  be  in  its  simplest  form  when  the 
smallest  possible  wfiole  nwnber  is  left  under  the  radical  sign. 


200,  I^rob,  2, —  To  simplify  a  radical^  or  reduce  it  to  its  lowest 
terms,  when  the  index  is  a  composite  number,  and  the  number  under 
the  radical  sign  is  a  perfect  poioer  of  the  degree  indicated  by  one  of 
the  factors  of  the  index. 

RULE. — Extract  that  root  of  the  number  which  corre- 
sponds TO  ONE  OF  THE  FACTORS  OF  THE  INDEX,  AND  WRITE  THIS 
ROOT  AS  A  SURD  OF  THE  DEGREE  OF  THE  OTHER  FACTOR  OF  THE 
GIVEN    INDEX. 

Dem. — The  mnXh.  root  is  one  of  the  mn  equal  factors  of  a  number.  If,  now, 
the  number  is  resolved  first  into  m  equal  factors,  and  then  one  of  these  m  factors 
is  again  resolved  into  n  other  equal  factors,  one  of  the  latter  is  the  mni\\  root  of 
the  number. 


201,  JProb,  3, — To  reduce  any  number  to  the  form  of  a  radical 
of  a  given  degree. 

RULE. — Raise  the  number  to  a  power  of  the  same  degree 
AS  the  radical,  and  place  this  power  under  the  radical  sign 

WITH    THE   REQUIRED   INDEX,  OR   INDICATE  THE    SAME   THING   BY  A 
FRACTIONAL   EXPONENT. 

Dem. — That  this  process  does  not  change  the  value  of  the  expression  is  evi- 
dent, since  the  number  is  first  involved  to  a  given  power,  and  then  the  corre» 
spending  root  of  this  power  is  indicated,  the  latter,  or  indicated  oi^ersiimn,  being 
just  the  reverse  of  the  former. 

202,  Cor. — To  introduce  the  coefficient  of  a  radical  under  the 
radical  sign,  it  is  necessary  to  raise  it  to  a  power  of  the  same  degree 
as  the  radical ;  for  the  coefficient  being  reduced  to  the  same  form  as 
the  radical  by  the  last  rule,  we  have  the  product  of  two  like  roots, 
which  is  equal  to  the  root  of  the  product. 


203,  Proh,  4, —  To  reduce  radicals  of  different  degrees  to  equiv- 
alent ones  having  a  common  index. 

RULE. — Express  the  numbers  by  means  of  fractional  in- 
dices. Reduce  the  indices  to  a  common  denominator.  Per- 
form UPON  THE  NUMBERS  THE   OPERATIONS  REPRESENTED  BY  THE 


CALCULUS   OF  RADICALS.  69 

NUMERATORS,  AND  INDICATE  THE  OPERATION  SIGNIFIED  BY  THE 
DENOMINATOR. 

Dem, — The  only  point  in  this  rule  needing  further  demonstration  is,  that  mul- 
tiplying numerator  and  denominator  of  a  fractional  index  by  the  same  number 

a  ma  n 

does  not  change  the  value  of  the  expression,  i.  e.,  that  x^  =  x"'".  Now,  x'^  signi- 
fies the  product  of  a  of  the  b  equal  factors  into  which  x  is  conceived  to  be  re- 
solved. If  we  now  resolve  each  of  these  b  equal  factors  into  m  equal  factors,  a 
of  them  will  include  ma  of  the  mb  equal  factors  into  which  x  is  conceived  to  be 
resolved.     Hence  ma  of  the  mb  equal  factors  of  x  equals  a  of  the  b  equal  factors. 

[The  student  should  notice  the  analogy  between  this  explanation  and  that 
usually  given  in  Arithmetic  for  reducing  fractions  to  equivalent  ones  having  a 
common  denominator.     It  is  not  an  identity.] 


204,  JProb,  S, —  To  reditce  a  fraction  having  a  monomial  radi- 
cal denominator,  or  a  monomial  radical  factor  in  its  denominator^ 
to  a  form  having  a  rational  denominator. 

RULE. — Multiply  both  terms  of  the  fraction  by  the  radi- 
cal IN  THE  DENOMINATOR  WITH  AN  INDEX  WHICH  ADDED  TO  THE 
GIVEN   INDEX   MAKES   IT   INTEGRAL. 


20S»  I^vob,  0. —  To  rationalize  the  denominator  of  a  fraction 
when  it  consists  of  a  binomial,  one  or  both  of  whose  terms  are  radi- 
cals of  the  second  degree. 

RULE. — Multiply  both  terms  of  the  fraction  by  the  de- 
nominator WITH  one  of  its  signs  CHANGED. 

Dem. — In  the  last  two  cases  the  student  should  be  able  to  show,  1st.  That 
the  operation  does  not  change  the  value  of  the  expression  ;  and,  2d.  That  it 
produces  the  required  form,  [This  is  the  substance  of  all  demonstrations  in  Me- 
ductions.] 


206,  I^rop.  1, — A  factor  may  be  found  which  will  rationalize 
any  binomial  radical, 

Dem. — If  the  binomial  radical  is  of  the  form  V(a  -F  fc)'",  or  {a  +  by ,  the  fac- 

n  —  m 

tor  is  (a  +  &)   "    ,  according  to  {204). 

L.        ]L  1 

If  the  binomial  is  of  the   form  "VaF  +  Vft*^,  or  «"'  +  6".     Let  a'"  =  x,  and 
1  LI 

b'^=y;  whence  a"*  =  a;* ,  and  b"*  =  y  ^    ^jgo  j^t  p  be  the  least  common  multiple 

up  rp 

of  m  and  n,  whence  x""  and  yp  are  rational.     But  x^  =  a"\  and  y"""  =  b".     If 
now  we  can  find  a  factor  which  will  render  x^  +  y^,  as»P  ±  y""",  this  will  be  a  fac- 


Vo 


LITEBAL  ARITHMETIC. 


L         L      1!L       ^ 
tor  which  will  render  a"  +  &*,   «"'±  6"  which  is  rational.    To  find  the  factor 

which  multiplied  by  x^  +  y"  gives  afp  ±  y'T,  we  have  only  to  divide  the  latter  by 
the  former.    Now ^—  =  x^p-^)  —  x^p-2)  «»•  +  x<p-*)  v^r  _a^(p-4)  yir    . 

;C*   +    y*"  ^  ^  if        T^ 

-  -  -  ±  y^p-')  (J),  the  +  sign  of  the  last  term  to  be  taken  when;?  is  odd,  and 
the  —  sign  when  it  is  even  (119).  Therefore  .t*<p-')  —  a'Kp-s)^'-  +  a^P-ny^r^ 
«^''~*V  + ±  y''^'"^,  is  a  factor  which  will  render  *'Va'+  Vibrational, 

x'  being  understood  to  be  a"*,  and  y  =  b* ,  and  p  tlie  L.  C.  M.  of  m  and  n. 

If  the  binomial  is  Va'  —  Vft^  ,  the  factor  is  found  in  a  similar  manner,  and  is 

af<-P-^)  +  x^P-^hf"  +  X<P-^*r  ^ ^  y^P-^), 


207.  Prop,  2. — A  trinomial  of  the  form  Va    +\/b    +Vc 
may  he  transformed  into  an  expression  with  but  one  radical  term  by 
midtiplyiiig  it  by  itself  with  one  of  the  signs  changed,  a»  \/  a  +  V  b 

—  V  c.  The  product  thus  arising  may  then  be  treated  as  a  binomial 
radical  by  coiundering  the  sum  of  the  rational  terms  as  07ie  term, 
and  the  radical  term  as  the  other. 

Thus,  {Va  +\/T  +  y/'c)  (y/a   +  \/~b—\/~c)  =  a  +  h—  c  +  2Vab.    Again, 
[(a  +  6  -  c)  -f-  2\/ab]  x  [(a  +  b  -  c)  ^  2v/a*]  =  a»  +  IP  +  c^  -  2ab  -  2bc 

—  2ac,  a  rational  result. 


Examples. 

1.  Reduce   the    following   to   their  simplest   forms:     VlOSa*x^^, 

j/2G46rAr«,    (7047a:i«t/«)i,    Vx^  -  x'^  +  a;*,     ^(x^-y^)    (x-y), 


{xr-^tr )  %     7A/363a;y3,     (a  -  b)  [{a^  -b^)(a-  b)Y,    5\rmx<^y^, 
ab\/Q3a-'xH-\ 

2.  Reduce  the  following  to  their  simplest  forms  (see  ScH.  199) : 

/!■  i/i-  \r\- 1/|-  fl-  V|.  D^T. 

li/?.    n/fs'    'fl-    i/f'    f?'-- 

x^  -  y^  .  /3a;g  +  6a;,y  +  dy^ 


CALCULUS  OF  RADICALS.  71 

3.  Reduce  the  following  to  their  simplest  forms  (see  200) : 

Vl^5a3^,  V363«8.c^  Vll^^^V,  V-1029a;i2, 
Vl35a*^  —  405rt3:c2  +  405rt22;3  _  I'dbax^. 

4.  Reduce  5fa*3  to  the  form  of  tlie  square  root;  also  7xt/;  also  J; 
also  Sa  —  2.     Reduce  2x^y^  to  the  form  of  the  3d  root, — to  the  form 

of  the  5th  root.     Reduce  -7-  to   the   form  of  the  4th  root, — to   the 

0 

form  of  the  cube  root.     Reduce  \^i  to  the  form  of  the  cube  root, — 
to  the  form  of  the  4th  root. 

5.  Introduce  under  the  radical  signs  the  coefficients  in  the  follow- 
ing expressions: 


2v^»  I  v/  3  ,  -iV  3  ,  i  V25,  2xVx^  (x  +  y)Vx^  -  3x^y  +  dxy^  -  y\ 
(x  +  y)V^^,     y  ^/Ubx,     «'(l--i)  • 

6.  Reduce  to  equivalent  forms  having  a  common  radical  index, 

\/2  and  \/3  >  ^^so  ^3,  Vo ;  also  \/2x,  \^3x^f  ^x,  and  ^/%x^  ;  also 

2\/c,    3v^,  and  jVo;  also  ^t's/hax,  2^2xy,  \/\^x\  also  a;  —  y  and 

{x  +  yy.     Explain  each  operation  upon  the  principles  of  factoring 
as  in  {203). 

7.  Prove  upon  the  principles  of  factoring  that  ^2  —  VS ;  also 
that  \/h  =  \/25 ;  also  that  \/3  =  ^21. 

8.  Reduce  the  following  to  equivalent  forms  having  rational  de- 

.     ^  2a\/5l'  5  ^/a        "s/  X       \^x  1  1 

nominators:  ■  ,     --7==,     —;=,      -tj=  ^    -Tr  ^     -J-y     "vh- ' 

V^x       2Va^      Vb       vy      Vy       V2       V2 

Jl_    VI    \^    \^ 

V'o'   V^'    </l'   'V3' 

9   Reduce  the  following  to  equivalent  forms  having  rational  de- 

.     ,  \/a;2  +  a:w  +  V*  ^^  ^  V^?^  —  Vy 

nominators: —  ,   — 3  ^   7=?     -^= ^r, 

V  a;  —  y  3  V  3  —  .t2     a;  +  V  y      yx  +  V  ?/ 

3  3  2  \/l2-  VlO        3  4-  2a/2 


VS  +  V2       a/3  +  A/5       V5  -  V4         a/6  +  a/5         A^S  -  V3 


72  LITERAL  AlilTHMETlO. 


Va^  +  :c2— a;*  Va:«  +  1  +  a;'  V^^^  -  V-^Tl' 

V.i'--  +  ^i:  +  1  +  Vx^  +  x-\  8  ^  3 

and  — =- 


Vic*  +  a;  +  1  -  Va:^  +  x  -  I    ^/'d  +\/^  +  l'  '        Vo  +  V3-a/3* 

10.  What  factor  will  rationalize  ^x—  ^yt    What  ^/Ic'^  —  ^/V^'i 
What  V8  + V3  + a/5? 

11.  By  what  must  numerator  and  denominator  of be 

multiplied  to  reduce  it  to  the  form  of  a  simple  fraction  ?    By  what 

Milt 

'     a;* 

12.  Introduce  the  coefficients  of  each  of  the  following  into  the 
parentheses:  8  (a*  —  x'^y,  a^(a  +  a^xy,  and  a;«(l  —  x^y. 


13.  Show  that  r  =  ;    also 

a  —  bx  -\-  Vn*  +  b^x^  bx 


that  ^-J!^  =  a/^^Lz21±. 


SECT/ON  IV. 
COMBINATIONS    OF    RADICALS. 


Addition  and  Subtraction. 
208,  IProh.  1. —  To  add  or  subtract  radicals. 

RULE.— It?  the  radicals  are  similar,  the  rules  already 
GIVEN  (66 f  71)  are  sufficient.    If  they  are  not  similar,  make 

THEM  SO  BY  (198^  203),  AND  COMBINE  AS  BEFORE.      If  THEY  CAN- 
NOT BE  MADE  SIMILAR,  THE  COMBINATIONS  CAN  ONLY  BE  INDICATED 

BY^  CONNECTING  THEM  WITH  THE  PROPER  SIGNS. 

[Note. — The  student  is  presumed  to  be  able  to  give  the  demonstrations  of 
the  problems  and  propositions  in  this  section,  as  they  are  but  a  recapitulation  of 
what  has  preceded.] 


CALCULUS  OF  RADICALS.  73 

Multiplication. 

209,  ^rop,  !• — The  product  of  the  same  root  of  two  or  more 
quantities^  equals  the  like  root  of  tlieir  ^woduct.     (See  104:*) 

210.  JProp,  2 •—Radicals  of  the  same  degree  arc  rniiltipUed  ly 
midtiplying  the  quantities  U7ider  the  radical  sign  and  wriiing  the 
product  under  the  common  sign. 

Similar  radicals  are  midtiplied  by  indicating  the  root  hy  fractional 
indices,  and,  for  the  product,  taking  the  common  number  with  an 
index  equal  to  the  sum  of  the  indices  of  the  factors.     (See  82,) 

211,  X^vob,  2, —  To  multiply  radicals. 

RULE. — If  the  factors  have  not  the  same  index,  reduce 

THEM  TO  A  COMMON  INDEX,  AND  THEN  MULTIPLY  THE  NUMBERS 
UNDER  THE  RADICAL  SIGN,  AND  WRITE  THE  PRODUCT  UNDER  THE 
COMMON  SIGN. 


Division. 


212,  I^rop, — TTie  quotient  of  the  same  root  of  two  quantities 
equals  the  like  root  of  tJieir  quotient. 

213,  JProb,.3, — 2h  divide  radicals. 

RULE. — If  the  radicals  are  of  the  same  degree,  divide 

THE  NUMBER  UNDER  THE  SIGN  IN  THE  DIVIDEND  BY  THAT  UNDER 
THE  SIGN  IN  THE  DIVISOR,  AND  AFFECT  THE  QUOTIENT  WITH  THE 
COMMON    RADICAL   SIGN. 

If  the    radicals   are  of  different   degrees,  reduce  THEM  TO 
THE   SAME    DEGREE   BEFORE   DIVIDING. 


Involution. 

214,  JProb,  4, —  To  raise  a  radical  to  any  power. 

RULE. — Involve  the  coefficient  to  the  required  power, 

AND  ALSO  THE   QUANTITY   UNDER  THE  RADICAL  SIGN,  WRITING   THE 
LATTER    UNDER  THE   GIVEN   SIGN. 

215,  Cor.  —  To  raise  a  radical  to  a  power  lohose  index  is  the  in* 
dex  of  the  root.,  is  simply  to  drop  the  radical  sign. 


74  LITERAL  ARITHMETIC. 

Evolution. 
2  to,  I^rob,  S, —  2h  extract  any  required  root  of  a  monomial 
radical. 
RULE. — Extract    the  required   root  of  the  coefficient, 

AND  OF  THE  QUANTITY  UNDER  THE  RADICAL  SIGN  SEPARATELY, 
AFFECTING  THE  LATTER  WITH  THE  GIVEN  RADICAL  SIGN.  REDUCE 
THE    RESULT   TO    ITS    SIMPLEST    FORM. 


[Note. — This  problem  should  not  be  taken  till  after  Quadratic  Equations.] 

217,  I^vob,  0» — To  extract  the  square  root  of  a  binomial,  one 
or  both  of  whose  terms  are  radicals  of  the  second  degree. 

Solution. — Such  binomials  have  either  the  form  a  ±  nVb  or  mV a  ±  nV b 
Now  observing  that  {x  ±  j/Y  =  x^  ±  2xy  +  y*,  we  see  that  if  we  can  separate 
either  term  of  any  such  binomial  surd  into  two  parts,  the  square  root  of  the  pro. 
duct  of  which  shall  be  \  the  other  term,  these  two  parts  may  be  made  the  first 
and  third  terms  of  a  trinomial  (corresponding  to  x^  ±  2xy  +  y^),  and  the  middle 
term  being  the  second  term  of  the  given  binomial,  the  square  root  will  be  the 
sum  or  difference  of  the  square  roots  of  the  parts  into  which  the  first  term  is 
separated. 

[Note. — This  process  requires  the  solution  of  a  quadratic  equation.  Thus  to 
extract  the  square  root  of  12  —  t^IiO.  Letting  x  and  y  represent  the  terms  of 
the  binomial  root,  we  have  «*  +  y*  =  12,  and  2xy  —  —  V'140.  Whence  x=  Vb 
or  VY,  and  y  —  V'7  or  V5,  and  the  root  is  V  o  —  V7.  The  sign  between  the  terms 
being  detennined  by  the  sign  of  the  surd  in  the  given  binomial.  On  this  ac- 
count this  subject  should  be  reserved  until  after  the  student  has  studied  quad- 
ratic equations,  or  the  solution  effected  by  inspection.  Thus,  in  this  example 
VliO  =  2^35.  Now  v'35  =  VS  x  Vl,  and  since  the  sum  of  the  squares  of  these 
factors  is  12,  we  have  Vl2  -  V140  =  \/E -  V7.] 


Examples. 

1.  Add  a/50  and  a/98.  Add  ^^n)tab^x  iind  \/^bUb^x.  Add 
^yiZTla*'X^  and  ^/bOOa^xK  Add  Vx^  and  a/«V-  ^"^^  A/n87j 
and -a/1008^  Add  5A/f  and  2\/^.  Add  a/|  and  ^^/U^.  Add 
y^l,  y^l  and  iv'sT    Add  ^/l^aH^,  VbOaJb^  and  dVlSaHxK 

2.  Show  that  a:y   1  +  (|)    -^yf   1  +  {^J=  {xi-hyh  (y*  +  a:^)* 

Show  that  iA^!^f^ 

y      a'+2ax  +  x^       J/     a^—'Zax  +  x^         \a^—x^J 


CALCULUS   OF   llADICALS.  75 


Add and ;  also  "^ ^ ^  ^ ^ and 

a  +  V  «^  —  2;2  a  —  V«^  —x^  Va  +  x  —  ya  —  x 

V(i  -{-x—Va- 


■x 


\/a  +  x+  "^  a—x 

3.  From  3  Vf  take  2VS-  F^'^^^i  ^/^^  take  v^24.  From  V^i 
take  Vf  From  f  Vf  take  -|A/f  From  \^a6^  take  ^^y .  From 
^'arb  take  aX^^ 


4.  Show  that  |/|i  - 1/3^  =  ttVe. 


5.  Show  that  A/"Vf:^':  -  a/^ 


-2rtZ>2  4_^3       4rt^,y<{, 


2ab  +  b^        a'^  —  b-' 

6.  Multiply  v^3  by  V^-*  Multiply  V^^"  by  v^a  Multiply 
VT  by  ^|,  V^2a^  by  \^ax^,  2\/^  by  3\^x^y,  a/1  +  ^  by 
\/rTx^,   vTby  -v/j,  2A/i  by  3a^2;  2^25  by  3V5,   v^24  by  e-^S, 

7.  Multiply  9  +  2\/r0  by  0  -  2\/l0,  v'^^  -  A^iiJ^  +  \/]f^  by 
\/x+\/y,  3\/5  +  2a/6-2  by  2^5  +  18^/0,  v^5-2v^6"by 
3^4-  v^36. 

8.  Divide  ^v^byiv^,  8\/9  by2\^,  a/6  by  v^,  i\/5by-jViO. 

9.  Divide  2 a/32  +  3  a/2"  +  4  by  4a/8^  4a/?  by  2a/^, 
6  +  2a/3  -  ^8  byA/6,  V^^^^a;  -  ^»2^-:r  by  Va'^,  /</  f"  by  /d/^, 
(rt  +  b)Va^^l-     by    («  -  Z>)  \/(a  +  1)2,    «  +  ^  -  c  +  2a/«^    by 

V^+ V^- V,I   I  !^l±^^  _  f^W^S  I  by  4|/-^. 
-    lx^-\/x^-a^      x'^  +  Vx^-a^  )  f    x^-i-a^ 

*  It  is  of  the  utmost  importance  that  the  pupil  be  able  to  give  a  complete  analysis  of  such 
examples.  Thus,  \/ 2  =  >/ S,  since  the  former  is  oneoi  the  two  equal  factors  of  2,  and  the 
latter  is  three  of  the  six  equal  factors  of  2.  In  like  manner VS  =  V 9.  Consequently 
v/2  X  V'3=  V^  X  ■n/Q.  Now  since  the  jjroduct  of  the  same  root  of  two  numbers  is  equal  to 
the  like  root  of  the  product,  VS  x  v/9  =  V72. 


76  LITERAL  ARITHMETIC. 

10.  Raise  3  v^2a;«  to  the  second  power.  Raise  ^  \/)lax^  to  the  5th 
power.      Cube    — fVf      Square  \/3  —  V^.     Cube  'dVa—x.     Cube 

11.  Extract  tlie  square  root  of  27  \^VS6x^y^  ;  the  cube  root  of 
,».riP*Vy;  the4throotof  25^*Vy;    the   5th   root  of  224  v^3^;   the 

cube  root  of  (1— a;)Vl— ^;    the  cube  root  of  vi/-;   the  square 

5  1/    X 

root  of  iVi- 

12.  Extract  the  squai:e  root  of  49+12a/5';  of  57  +  12a/15";  of 
{a^-ha)x-2axVa;  x-2\/^x^;  of  VI8-4;  of -^+-|Va«- c«. 
(See  ;^ir.) 

SECT/ON    F. 

IMAGINARY    QUANTITIES. 

)^18.  ^n  Imaginary  Quantity  is  an  indicated  even  root 
of  a  negative  quantity,  or  any  expression,  taken  as  a  whole,  which 
contains  such  a  form  either  as  a  factor  or  a  term. 

Thus  >/ —x,\/ —y* ,  S^—x*,  24^^^— 4,  V— 6,  3— V—1,  etc.,  are  imaginary 
quantities. 

219,  ScH.  1. — It  is  a  mistake  to  suppose  that  such  expressions  are  in  any 
proper  sense  more  unreal  than  other  symbols.  The  term  Impossible  Quantities 
should  not  be  applied  to  them :  it  conveys  a  wrong  impression.  The  ques- 
tion is  not  whetlier  the  symbols  are  symbols  of  real  or  unreal  (imaginary) 
quantities  or  operations,  but  what  interpretation  to  put  upon  them,  and  how 
to  operate  with  them  when  they  occur. 

220.  Sen.  2. — A  curious  property  of  these  symbols,  and  one  which  for 
some  time  puzzled  mathematicians,  appears   when  we  attempt  to  multiply 

y/—xhY\/—X'  Now  the  square  root  of  any  quantity  multiplied  by  itself 
should,  by  definition,  be  the  quantity  itself  ;  hence  V—xx  \/—x=  —x. 
But  if  w^e  apply  the  process  of  multiplying  the  quantities  under  the  radicals, 
we  have  v/^  x\/^  =Vx^  =  +  a;  as  well  as  —  x.  What  then  is  the  pro- 
duct of  v/3^  W^^  ?  Is  it  —a-,  or  is  it  both  +  x  and  —x  ?  The  true 
product  is  —x  ;  and  the  explanation  is,  that  v/a;*  is,  in  general.,  +x  and  —x. 
But  when  we  know  what  factors  were  multiplied  together  to  produce  a;*,  and 
the  nature  of  our  discussion  limits  us  to  these,  the  sign  of  \/x^  is  no  longer 
ambiguous  :  it  is  the  same  as  was  its  root. 


CALCULUS   OF   IMAGINAKIES.  77 

221.  Prop. — Every  imaginary  monomial  can  be  reduced  to  the 

form  ni  V  —  J^j  ^'^  which  m  is  real  (not  imaginary),  m  may  be 
rational  or  surd. 

p 

Dem.     X  V  —y,  p  being  an  even  number,  is  the  general  symbol  for  an  imagin- 
ary monomial.     Now  if  p  is  a  power  of  2,  we  may  write  at  once  p  =  2»,  whence 

p 2" 2" 2" 2" 2" - 

.f  V  —  i/  =  :v  V—i/  —  X  V(/{~1)  =x  Vy     V—  1  =  m  V—1.    If  p  contains  other 

factors  than  2,  let  r  represent  their  product,  and  2»  the  product  of  all  the  factors 

of  2  contained  in  p  ;  whence  p  =  r2",  in  which  r  is  odd,  since  the  product  of  any 

p. r2*!^ rS" 

number  of  odd  factors  is  odd.  We  then  have  x  V—y  =  x  V —y  =  x  Vy{—V) 
z=  X  V  y  V  —  1  —  X  yy  y  V—1  =  x  Vy  V—1,  since  any  odd  root  of  —  1 
is  —  1.      Putting   X  V  y  =  m,  this  becomes  m  V—1- 

222,  ScH. — When  n—1^  i.  e.,  when  there  is  but  one  factor  of  2  in  the 
index  of  the  root,  the  form  becomes  m  V—1.      This  form  is  called  an  imagin- 

ary  of  the  second  degree  ;  m  r  —  1  is  of  the  fourth  degree,  etc.  In  this  dis- 
cussion we  shall  confine  our  attention  mainly  to  imaginaries  of  the  second 
degree. 


223,  I^vob, — To  add  and  subtract  imaginary  mo7iomials  of  the 
second  degree^  or  such  as  may  be  reduced  to  this  degree. 

RULE. — Reduce  them  to  the  form  mV—  1?  and  then  com- 
bine   THEM,    CONSIDERING    THE    SYMBOL   V—  1   AS    A    SYMBOL    OF 

character.* 

Examples. 
1.  Add  \/^=T  and  ^/~^. 

Operation.     V~^4:  =  V^{  -  1)  =  2  V^^,  and  V^^  =  3  V^. 
.-.    V^^4:  +   V^^9  =  2  V^l  +  3  V^^l  =  5  V^^l. 

Sen. — The  last  operation  should  not  be  looked  upon  as  taking  the  sum  of 
2  times  a  certain  quantity  (represented  by  \/—l)  and  3  times  the  same 
quantity,  but  ns  2  of  a  certain  character  added  to  3  of  the  same  character. 

■*  St-e  (■!*,  49 f  50).  Wc  mean  to  pay  that,  as  a  qnantity  (considered  numerically),  m  and 
m  V—  1,  are  exactly  tlie  same,  just  as  is  the  case  in  the  expreseions  +  m  and  —  m ;  but  that  the 
eynibol  $/  —  i  gives*  gome  peculiar  or  concrete  significance  to  m,  as  does  the  sign  +,  or  — ,  or  f. 
What,  this  concrete  pigniflcance  is,  we  cannot  here  say.  It  has  its  clearest  interpretation  in  the 
Co-ordinate,  or  General  Geometry. 


78 


LITERAL  ARITHMBTia 


Thus  the  operation  of  reducing  ^/ —^  and  ^ —^Xa  the  forms  %^^\  and 
3^/— 1  is  to  be  looked  upon  as  rendering  the  expressions  homogeneous.  It 
is,  however,  to  be  observed  that  the  symbol  ^ —\  is  subject,  also,  to  the 
ordinary  laws  of  number,  and  may  be  operated  upon  accordingly.  Thus  it 
has  a  double  significance. 

2.  Add  4V  -%n  and  3V  -  16;  also  ^a^^^lib  and  %a^/~^^', 
also  ^V^^^  and  c\/"^.     V^f^^  +  cv/^^  =  {W  +  3c)V^. 

3.  Add  V-  1024  and  V-729. 

Opkration.     ^-1024  =  ^i^  \/^'  and  ^-729=  ^729  aJ~^\. 
.-.    ^^^=^024  +  >v/-  729  =  59 v^^. 

4.  Show  that  in  general  v^  —  a;  ifc  "s/  —  y  =  {\/ x  dc  \^)V^  —  1, 
■when  ^t  is  any  integer. 

5.  From  V^^  take  \/"^4.  From  4a/^=^7  take  3V^^6. 
From   3fl\/  —  25    take  2rt V  —  4.      Show  that  aV  —  6  —  cV  —  d 

6.  From  V-  ^090  take  V  -  9.  i^cw.  ClV— "L 

8cH. — It  would  seem  improper  to  onrft  the  1  before  the  symbol  /y/— 1  in 
such  a  case  as  the  last,  thougli  it  has  been  customary  to  do  so.  If  we  arc  to 
consider  \/— 1  as  a  sign  of  character  ("affection,"  as  some  say),  there  is  no 
more  reason  for  omitting  the  1  in  such  a  case,  than  there  is  in  such  as  4  —  5 
=  —  1.  That  is,  if  we  write  4^^/^  —  S^y/^  =  'y/^,  we  ought  to  write 
4  —  5  =  —,  or  $5  —  $4  =  I,  to  be  consistent. 


7.  Show  that  V^S  +  V~-^l^  =  5  V^^  =  bV^V  -  1 ;  also 
that  4:\/~^^  4-  2V'^^^^n  =  16 a/3  V^^  ;  also  that  a/~^=1g 
—  \/  —  1  =  1  \/  —  1  ;     also  that  2\/  —  a  —  \/  —  a  =  \^\^  —  1. 


224,  JPt*op, — Evert/  polynomial  coiitainiiig  some  real,  and 
some  imaginary  terms  of  the  second  degree,  or  such  as  can  he  re- 
duced to  this  degree,  can  he  reduced  to  the  form  adbbv  —  1»  **'* 
which  a  and  b  are  real,     a  and  b  may  he  rational  or  surd. 

Dem, — This  is  evident  from  the  fact  that  all  the  real  terms  can  be  combined 
into  one  (it  may  be  a  polynomial)  and  represented  by  a,  and  the  imaginary  terms 
being  reduced  to  the  form  m  V^  1  can  also  be  combined  into  one  term  repre- 
sented by  ±  &  V^. 


CALCULUS   OF   IiMAGINARIES.  79 

225,  ScH. — The  form  a±h^  —  l  is  considered  the  gcncnil  form  of  an 
imaginary  quantity  of  the  second  degree.  Wlien  «  =  0,  it  becomes  tlie  samo 
as  m^/—l.  The  two  expressions  a  +  h\^—l  and  a  —  h\/  —  l  arc  called  Con- 
jugate Imaghmries.  Hence  tlie  sum  of  two  conjugate  imaginaries  is  rcil 
(3a).     Also  the  product  of  two  conjugate  imaginaries  is  real  [{a  +  h\/  —  \) 

X  (a  —  J^— 1)  =:  a'^  +  Jr^  as  will  appear  hereafter].  The  square  root  of  the 
product  of  two  conjugate  imaginaries,  taken  with  the  4-  sign,  is  called  the 
Modulus  of  each.  Thus  +  /y/^"^  +  b^  is  the  modulus  of  both  a  +  'b^^  —I  and 
a  -  h^/^. 

Exs. — Fi lid  the  sum,  and  the  difference  of  2  +  V^  and  3  —  V— 04  ; 
also  of  a  +  \/  —  d  and  a+V—o.     Last  results^  2a  +  ( V^  +  Vc)  V'  —  i , 


Multiplication  and  Involution. 

220,  I* rob, —  7b  determine  the  character  of  the  product  of  sev- 
eral imaginary  monomiaX  factors  of  the  second  degree. 

Solution. — Letting  n  represent  any  integer,  including  0,  47i,  4w4-l,  4n  +  2, 
and  4/2 +  3  may  represent  all  integral  exponents,  so  that  all  the  forms  to  be  treated 

are  (^"1^,  (V^)'"^',  (^=1)"''".  ^"^  W~^T'" -  Thus,  if  n  =  0. 
4//  +1  =:  1,  4/1  +  2  =  2,  and  4n4-3  -  3.  If  n  =  1,  4m  =  4,  4r.+  1  =  G,  4;^  +  :?  =  C, 
and  4n  +  3  =  7.    If  n  =  2,  4n  =  8,   etc. 

We  shall  now  show  that         (V~0^'      =  1» 

_  and       _^    (.y/3i)*^+' ^  _^Z:i. 

(v^ir  =  ( v^i  v^i  ^  v^i  v^i)"  =  [(-1)  X  (-1)]"  =  i«  =  1, 

since    y'— 1  ^/^l  =  —1   by  {220\   (—1)  (—1)  =  1,  and  any  power  of  1  is  1. 
(a/~1)'*^'  =  (V=ir  X  V^l  -  V=l  =  a/-1.    since   (y— 1^  =  1. 

(V=i)"^'  =  (V~i)"  X  ( V-iy  =  1 X  (-1)  =  -1. 

(y^— 1)^-+^  ^  {^~\)'^^   X    V-^1  =  (-1)  V=^  =  -  V=T  =   -  V~,. 

III. — To  find  what  (-\/— l)*  is,  we  have  but  to  observe  that  if  7i=l,  471  +  1=5, 

and   (vri/  =  (y— l)^«^^y'i:i.      ^ 

Again,    (-y/^)'  =  (V-^^""'  =  -  a/-^.  ^i^^cc,  if  7i  =  0,  4/i  +  3  =  3. 

Once  more,    {\^\Y^  =  {\^~\Y"^'^  =  —1,  since,  if  ti  =  2,  4n  +  2  =  10. 

In  like  manner,    {^/—\Y  =  {\^—l)  "  =  1,  since,  if  n  =  1,  4//.  =  4. 

SCH. — In  the  above  we  have  confined  ourselves  to  the  powers  of  the  posi- 
tive square  root,  sinca  —  \/ — 1  may  be  understood  to  be  —  1 V  —  ^ '  ^^-^  ^^^^  factors 
-1  bo  treated  separately.  Thus,  (-  ^/'-^Y"  r=  (-l)'"  (^/-lY"  =  (  V"'  )** 
=-  1.  So    also    (-  V~0'"''  =  (-1)*"'  X   (V-iy-i  =  (-1)a/-1  = 


So  LITERAL   ARITHMETIC. 

—  \/^^,  etc.     Or,  in  other  words,  giving  the  —  sign  to  ^/—l  simplj  changes 
the  signs  of  the  odd  powers.     In  examples  we  shall  confine  attention  to  the 

+  root  of  ^Z  — 1. 

Examples. 

1.  Multiply  4\/^=^  by  2V~^^. 

Operation.    4  V^  =  4  Va  V^,  and  2  V^  =  2  V2"  V^. 

.'.  4  V^  X  2  V^  =aVs  V^  X  2  V2  V^  =  8  V^{^^^=  -SVQ. 

2.  Show  Unit  V—x^  X  \/  —  7j^  =  —x)/;  also  tliatSV—  5x4^—  3 
=  —12  Vl^     What  is  the  product  of  ~2V~-^  by  -3  V^^  ? 


3.  Show  that  V—  2  x  V—  8  =  W—  1;  also  that  V—  256  x 
V'^^27  =  48\/3  V^^^;  also  that  \/^^  x  W'^^  =  W^  V^- 

4.  Show   that  4a/— T  +  V— ^  multiplied  by  2a/—  1  —  V—  3 
equals  a/O  +  4a/3  —  2\/2  —  8. 

5.  Show  that  J  —  iv^^^  squared  equals  —  J(l  +  V—  3). 


6.  Showthat(3-2A/^^)x(5  +  3A/-  4)=39-2a/-  1 ;  also  that 

(i+v^=nr)x(i-A/'^=n:)=2. 

7.  Sliow  tliat  2  is  the  modulus  of  V2 -\-  V~^^  and  a/2  —  a/^^- 
What  is  the  modulus  on]i-2V'^^>  and  3-2a/^^?  Ofo-SV-i^ 

8.  Show  that  (a/^)''i=  _  75a/^=3;  also  that  (V^^Y^^S^ 
0.  AVhat  is  the  5th  power  of  2V~-^?    Of  3  a/^^^  ? 


10.  What  is  the  product  of  a/— a;^,  a/-^/-,  a/— ^S  and  V—w^? 


Division"  of  Imaginaries. 

227,  I* rob. —  To  divide  one  imaginary  of  the  second  degree  by 
another. 

RULE. — Reduce  tfie  imaginary  term,  or  terms,  to  the  form 

m\/  —  1.  or  mW  —  l)",  AND  DIVIDE  AS  IN  DIVISION  OF  RADICALS, 
OBSERVIXG  THE  PRINCIPLES  OF  {220)  TO  DETERMINE  THE  CHARAC- 
TER OF  THE  QUOTIENT  OF  IM  AG  IN  ARIES. 


CALCULUS   OF  IMAGINARIES.  81 

Examples. 


1.  Divide  \/-  16  by  V  -  4. 

Operation.  V—  16  =  41^^,  and  V^^  =  2V^;  .-.  V^^^  -^  V^IIl 
=  44/31  -^  2f^^  =  2(  V=l)°.  But  by  (226)  {V^^lY  =  +  1 ;  hence  the 
quotient  is  2. 

ScH. — A  superficial  view  of  the  case  might  make  the  quotient  ±  2.  Thus, 
as  the  radicals  are  similar  it  might  be  inferred  that  V^l  -^  V^^  =  4/  3_ 
=  \^  =  ±1.      (See  220,) 

2.  Show  that  6V  -  3  -4-  2 V"^^  =  iV~S ;  also  that  -  V~^^ 
-^  -  6\/~^^  =  t^a/3;  also  that  1  -^  V  -  1  =  -  V"^ ;  also 
that  6  ^  2^^^  =  -  3  V^^. 


3.  Divide  2^/- 1  by  V -2;  also  3V  -  16  by  -  12;  also  Va 
by  V^^' 

SuG's.  2 v^  -^  V:i2  =  V 16  VFi?  -  V^  V^i  =  V^  V^. 

4.  Show  that  sV^^H:^  -^  2  v""^^  =  4v^2'>/^=l. 

5.  Show  that  (1  +  V"^^)  -^  (1  -  V"^^)  =  V^^;  also  that 
(4  4.  ^/~Ir2)   -^   (2   -  V"^^)     =    1    +    V^a/""^  ;     also     that 


1  ^  (3  _  2V  -  3)  =  — ^^^ — - — ;   also  that  1  -4 


21  a  +  V  -  .^• 

a^  —  X  -\-  2aV  —  X.  ,       ^1    i.    ^^  +  V  —  ^      ,      a  —  V  —  0 

: ■  >     also   that    7==     +     y= 

a'^  +  X  a  —  ^/  —  h            a  ■\-  \  —h 

2ia^  -  b) 

''     a^  +  b   ' 

6.  Simplify 


{a  +  bV  -if  +  (g-^V-l)^ 


[Note. — Here  ends  the  subject  of  Literal  Arithmetic.  The  student  is  now 
prepared  for  the  study  of  Algebra,  properly  so-called ;  i.  e.,  The  Science  of  the 
Equation.] 


PART    IL 


AN    ELEMENTARY    COURSE    IN 
ALGEBRA. 


OHAPTEE  I. 

SIMPLE   EQUATIONS. 


SECT/ON  L 

EQUATIONS  WITH  ONE  UNKNOWN  QUANTITY. 

Definitions. 

1.  An  Equation  is  an  expression  in  mathematical  symbols,  of 
equality  between  two  nnmbers  or  sets  of  numbers. 

2.  Algebra  is  that  branch  of  Pure  Mathematics  wliich  treats 
of  the  nature  and  properties  of  the  Equation  and  of  its  use  as  an 
instrument  for  conducting  mathematical  investigations. 

3.  The  First  Member  of  an  equation  is  the  part  on  the  left 
hand  of  the  sign  of  equality.  The  Second  Member  is  the  part 
on  the  right. 

4.  A  ^tnnerical  Equation  is  one  in  which  the  knoivn 
quantities  are»represented  by  decimal  numbers. 

5.  A  Literal  Equation  is  one  in  which  some  or  all  of  the 
known  quantities  are  represented  by  letters. 

6.  The  Degree  of  an  Equation  is  determined  by  the  liighest. 
number  of  unknown  factors  occurring  in  any  term,  the  equation 
being  freed  of  fractional  or  negative  exponents,  as  affecting  the  un- 
known quantity. 

7.  A  Simple  Equation  is  an  equation  of  the  first  degree. 

8.  A  Quadratic  Equation  is  an  equation  of  the  second 
degree. 

9.  A  Cubic  Equation  is  an  equation  of  the  third  degree. 


TRANSFORMATION     OF  EQUATIONS.  8^' 

10.  Equations    above    the    second  degree  are  called   Higher 

Equations,     Those  of  the  fourth  degree  are  sometimes  called 
Biquadratics. 


Transformation  of  Equations. 

11,  To  Transform  an  equation  is  to  change  its  form  without 
destroying  the  equality  of  the  members. 

12,  There  are/owr  principal  transformations  of  simple  equations 
containing  one  unknown  quantity,  viz :  Clearing  of  Fractions,  Trans- 
position, Collecting  Terms,  and  Dividing  by  the  coefficient  of  the 
unknown  quantity. 

13,  These  transformations  are  based  upon  the  following 

Axioms. 

Axiom  1. — Any  operation  may  be  jyerfornted  upon  any  term  or 
upon  either  member^  which  does  not  affect  the  value  of  that  term  or 
member^  without  destroying  the  equation. 

Axiom  2. — If  both  members  of  an  equation  are  increased  or  di- 
rninished  alike,  the  equality  is  not  destroyed. 


14,  JProb, —  To  clear  an  equation  of  fractions. 

RULE. — Multiply  both  members  by  the  least  or  lowest 

COMMON  MULTIPLE  OF  ALL  THE  DENOMINATORS. 

Dem. — This  process  clears  the  equation  of  fractions,  since,  in  the  process  of 
multiplying  any  particular  fractional  term,  its  denominator  is  one  of  the  factors 
of  the  L.  C.  M.  by  which  we  are  multiplying ;  hence  dropping  the  denominator 
multiplies  by  this  factor,  and  then  this  product  (the  numerator)  is  multiplied  by 
the  other  factor  of  the  L.  C.  M. 

This  process  does  not  destroy  the  equation,  since  both  members  are  increased 
or  diminished  alike. 

III. — An  equation  is  aptly  compared  to  a  pair  of  scales  with  equal  arms,  kept 
in  balance  by  weights  in  the  two  pans. 


Transposition. 

IS,  Transposing  a  term  is  clianging  it  from  one  member  of 
the  equation  to  the  other  without  destroying  the  equality  of  the 
members. 


64  ELEMENTARY  ALGEBRA. 

16,  Frob, — To  tra?ispose  a  term, 

RULE.— DllOV  IT  FROM  THE  MEMBER  IN  WHICH  IT  STANDS  AKD 
INSERT   IT   IN   THE   OTHER   MEMBER   WITH   THE   SIGN   CHANGED. 

Dem, — If  the  tenn  to  be  transposed  is  +,  dropping  it  from  one  membcir 
diminishes  that  member  by  the  amount  of  the  term,  and  writing  it  with  the  — 
sign  in  the  other  member,  takes  its  amount  from  that  member;  hence  both 
members  are  diminished  alike,  and  the  equality  is  not  destroyed.  (Repeat 
Axiom  2.) 

2d.  If  the  term  to  be  transposed  is  — ,  dropping  it  increases  the  member  from 
Avhich  it  is  dropped,  and  writing  it  in  the  other  member  with  the  +  sign  i7i- 
creases  that  member  by  the  same  amount ;  and  hence  the  equality  is  preserved. 
(Repeat  Axiom  2.) 

17,  To  Solve  an  eqiuitioii  is  to  find  the  value  of  tlie  unknown 
quantity;  that  is,  to  find  what  yaliie  it  must  have  in  order  that  the 
equation  be  true. 

18,  An  equation  is  said  to  be  Satisfied  for  a  value  of  tlie  un- 
known quantity  which  makes  it  a  true  equation ;  i.  e.,  which  makes 
its  members  equal. 

19,  To  Verify  an  equation  is  to  substitute  the  supposed  value 
of  the  unknown  quantity  and  thus  see  if  it  satisfies  the  equation. 

Sen.  2. — The  pupil  must  not  understand  that  the  verili cation  is  at  all 
necessary  to  prove  that  the  value  found  is  the  correct  one.  This  is  demon- 
strated as  Are  go  along,  in  obtaining  it.  Tlie  object  of  the  verification  is  to 
give  the  pupil  a  clearer  idea  of  the  meaning  of  an  equation,  and  to  detect 
errors  in  the  work. 


20,  Proh,  1, —  To  solve  a  simple  equation  with  one  unknown 
quantity. 

R  TILE. — 1.  If  the  equation  contains  fractions,  clear  it  of 
THEM  BY  Art.  14. 

2.  Transpose  all  the  terms  involving  the  unknown  quan- 
tity to  the  first  member,  and  the  known  terms  to  the  second 
member  by  Art.  16. 

3.  Unite  all  the  terms  containing  the  unknown  quantity 
into  one  by  addition,  and  put  the  second  member  into  its 
simplest  form. 

4.  Divide  both  members  by  the  coefficient  of  the  unknown 
quantity. 


SOLUTION  OF  SIMPLE  EQUATIONS.  86 

Dem. — The  first  step,  clearing  of  fractions,  does  not  destroy  the  equation, 
since  both  members  are  multiplied  by  the  same  quantity  ^^AxiOM  2). 

The  second  step  does  not  destroy  the  equation,  since  it  is  adding  the  same 
quantity  to  both  members,  or  subtracting  tiie  same  quantity  from  both  members 
(Axiom  2). 

The  third  step  does  not  destroy  the  equation,  since  it  does  not  change  the 
value  of  the  members  (Axiom  1). 

The  fourth  step  does  not  destroy  the  equation,  since  it  is  dividing  both  mem 
bers  by  the  same  quantity,  and  thus  changes  the  members  alike  (Axiom  2). 

Hence,  after  these  several  processes,  we  still  have  a  true  equation.  But  now 
the  first  member  is  simply  the  unknown  quantity,  and  the  second  member  is  all 
known.    Thus  we  have  ichat  the  unknoicii  quantity  is  equal  to  ;  i.  e.  its  value. 

21,  Sen.  1. — It  must  1)6  fixed  in  the  pupiVs  mind  that  lie  can  make  htt  two 
■classes  of  chwiges  upon  an  equation:  viz.,  Such  as  do  not  affect  the  value 
OF  the  members,  or  such  as  affect  both  members  equally.  Every  opera- 
tion must  be  seen  to  conform  to  these  conditions. 

22,  Cor.  1. — All  the  si(/ns  of  the  tenuis  of  both  members  of  an 
equation  can  be  changed  from  +  to  —,or  vice  versa,  without  destroy- 
ing the  equality.^  since  this  is  equivalent  to  midtiplying  or  dividing 
by  -1. 

23,  ScH.  2. — It  is  not  always  expedient  to  perforin  the  several  trans- 
formations in  the  same  order  as  given  in  the  rule.  The  pupil  should  bear  in 
mind  that  the  ultimate  object  is  to  so  transform  the  equation  that  the  un- 
known quantity  will  stand  alone  in  the  first  member,  taking  care  that,  in 
doing  it,  nothing  is  done  which  will  destroy  the  equality  of  the  members. 

24,  ScH.  3. — It  often  happens  that  an  equation  which  involves  the  second 
or  even  higher  powers  of  the  unknown  quantity  is  still,  virtually,  a  simple 
equation,  since  these  terms  destroy  each  other  in  the  reduction. 


Simple  Equation's  containing  Radicals. 

25,  Many  equations  containing  radicals  which  involve  the  un- 
known quantity,  though  not  primarily  appearing  as  simple  equations, 
become  so  after  beinsr  freed  of  such  radicals. 

26,  JProb,  2, —  To  free  an  equation  of  radicals. 

RULE. — The  common  method  is  so  to  transpose  the  terms 

THAT  the  radical,  IF  THERE  IS  BUT  ONE,  OR  THE  MORE  COMPLEX 
radical,  IF  THERE  ARE  SEVERAL,  SHALL  CONSTITUTE  ONE  MEMBER, 


86  ELEMENTARY  ALGEBRA. 

AND  THEN  INVOLVE  EACH  MEMBER  OF  THE  EQUATION  TO  A  POWER 
OF  THE  SAME  DEGREE  AS  THE  RADICAL.  If  A  RADICAL  STILL  RE- 
MAINS, REPEAT  THE  PROCESS,  BEING  CAREFUL  TO  KEEP  THE  MEM- 
BERS  IN  THE   MOST   CONDENSED   FORM   AND   LOWEST  TERMS. 

Dem. — That  this  process  frees  the  equation  of  the  radical  whicli  constitutes 
one  of  its  members  is  evident  from  the  fact  that  a  radical  quantity  is  involved 
to  a  power  of  the  same  degree  as  its  indicated  root  by  dropping  the  root  sign. 

That  the  process  does  not  destroy  the  equality  of  the  members  is  evident  from 
the  fact  that  the  like  powers  of  equal  quantities  are  equal.  Both  members  are 
increased  or  decreased  alike. 


Summary  of  Practical  Suggestions. 

27,  In  attempting  to  solve  a  simple  equation,  always  consider, 

1.  Whetlier  it  is  best  to  clear  of  fractions  first 

2.  Zook  out  for  compound  negative  terms. 

3.  If  the  nnmerators  are  polynomials  and  the  denominators  mono- 
mials, it  is  often  better  to  separate  the  fractions  into  parts. 

4.  It  is  often  expedient,  when  some  of  the  denominators  are  mono- 
mial or  simple,  and  others  polynomial  or  more  complex,  to  clear  of 
the  most  simple  first,  and  after  eacli  step  see  that  by  transposition, 
uniting  terms,  etc.,  the  equation  is  kept  in  as  simple  a  form  as  pos- 
sible. 

5.  It  is  sometimes  best  to  transpose  and  unite  some  of  the  terms 
before  clearing  of  fractions. 

G.  Be  constantly  on  the  lookout  for  a  factor  which  can  be  divided 
out  of  both  members  of  the  equation,  or  for  terms  Avhich  destroy 
cacli  other. 

7.  It  sometimes  happens  that  by  reducing  fractions  to  mixed  num- 
bers the  terms  will  unite  or  destroy  each  other,  especially  when  there 
are  several  polynomial  denominators. 

28.  When  the  equation  contains  radicals,  specially 
consider, 

1.  If  there  is  but  one  radical,  by  causing  it  to  constitute  one  mem- 
ber and  the  rational  terms  the  other,  the  equation  can  be  freed  by 
involving  both  members  to  the  power  denoted  by  the  index  of  the 
radical. 


SOLUTION   OF  SIMPLE  EQUATIONS.  8? 

2.  If  there  are  two  radicals  and  other  terms,  make  the  more  com- 
plex radical  constitute  one  me!uber,  alone,  before  squaring.  Such 
cases  usually  require  two  involutions. 

3.  If  there  is  a  radical  denominator,  and  radicals  of  a  similar  form 
occur  in  the  numerators  or  constitute  other  terms,  it  may  be  best  to 
clear  effractions  first,  either  in  whole  or  part. 

4.  It  is  sometimes  best  to  rationalize  a  radical  denominator. 


Examples  for  Practice  in  Solving  Simple  Equatioists. 

1.  Solve  and  verify  the  following :        (1.)  40— 6a:— 16=120— l4;r. 

X       X     x_  a:  -  3      a;  _  ^^'-19       /.w^+S    a; 

<^-)  2~3'^4-^^-     (^'^  "2~'^3-^^~^-      (^•)"T~'^3 

.     «— 5       /^x    9.T  +  20       4a:— 12       x       ,„.    lOic  +  17        12a:  +  2 

5a:— 4      ,^.   ax—h      a      hx      hx—a       ,_ .    aih^+x^)  cix 

(9.)  3:2^3  =  ax  +  bx  +  ex.       (10.)    2.04  -  0.68y  -  0.02y  =  0.01. 
(11.)  8.4a:  -  7.6  =  10  +  2.2a:. 


o    o  1         /i  \        1        ,        1  1  tc,\    AQ        .'i'2a:-.05 

2.  Solve     (1.)  -T \- 1 r-  — •     (^O   4:.8a; — 

^    '  ah— ax      be— ox       ac—ax      ^    '  .5 

=  1.6^  +  8.9.    (3.)  ^±^'^11^  =  li^lZ^.  (x  =  ^ifc^).     (4.)^ 
^    '       p—q  m        \  m    J      ^    '  2h—a 

(Sbc  +  mVjx  _    bah    _  {Uc-a(l)x  _  6a(2b—a)    /  _  5^h-a)\ 
~  2ah(a  +  h)   ~  3c^ "  2ab(a-h)  ^^~h^'  V~      'Sc-d    J' 

ax       ^j  _     ex       (   _^ah^+4.h^~Ua^b\  Am(K^-bx') 

^""'^  ^{Zr^'^'^^-^da  +  b'  V"  '6a^+ab-ac  +  be)'     ^^^  %x 

-imp^  4         '   V-^^p^-bg^y     ^^'Hx^dx^fx^hx-"' 

8.5       .2  1  -  .Ix  2 -3a:       5a:      2a:-3_a:-2 


(10.)  5-a:(3i-?)=ia:- 


3a:  -  (4  -  5a:) 


^  ELEMENTARY  ALGEBRA. 

3.  In  solving  the  following  be  careful  to  observe  the  suggestions 
in,.,,:    (..,|H)-JH)*K-~i)-»-    »•¥-' 

_x—5      x—6 
~  x—(j      x—7' 

4.  Solve  the  following,  giving  special  heed  to  the  suggestions  in 

^ ^  it 

(28) :      (1.)  V^^=^  =  16  -  V^.      (2.)     ^  —  =  6-.       (3.)  Vx 

Vb^  +x—b 

-^Va;  — 7= — .     (4.)  — ==::= 1  = .     (o.)    ^ a-VV  x 

\^x-l  \/5x4-3  2 

+  Vrt_V^=V^.        (6.)    V(l+rt)*  +  (l-a)a:4-V(l-«)'  +  (l+rt)i 


=2fl.      (7.)  V!l3  +  V'[7  +  A/(3  +  V.r)]!=4.     (8.)  ^^^lTV(3 -f  V6a.-) 

...      (9.)  Vjng^^VO^-Q^      (10.)  ^^^+^^\^=1G.-8V3^ 
V(3;c  +  2     4V««+6  16a;-3 

+  3.     (11.)    ^^-^^"^^^  =  ^.     (12.)    "f_-^     =4  +   ^^'-' 


f?  4-  V«*  —  X*  's/ax  +1  ^ 

nq\     3V^— 4  _  15  +  \/9^  a/^  +  V^  _  Va  4-  a/^ 

2  +  a/^         40  +  a/^  '      V«^  —  V^  V^ 


/i-\      a/«  — '^rt— a/<^*  —  «a;        .,       ,.^.   V in  ^  \/ m  —  y       1 

(lo).       =r =   0,        (lb.)     — — = ;z=^   = — • 

^/liJ^W  a-^ni^-ax  V  m  -  ^/ m  -  y      ^ 


(17.)     ^  +  ^+V'j^tf!  =  J.  (18.)    ^^^^-^^-^  =  i. 

rt  +  .T  —  A/2rta;  +  x^  yx  +  1  +  V  a;  —  1 


/^ci\  J/~~r-  .    .V 7.     ,^_  .  l+a;+ a/2^+^         a/^  +  sj+a/^ 

(19.)  A/rt  +  .y  +  Va—x  =  b.    (20.) =rt — — . 

1+x—  V^'^x  +  x^         V^-hx—Vx 

(^l.)^S±^=4.      (22.)   -^=3 -  +  J -. 

\/ox-\-l—\^'6x  Va  —  x  +  Va       ^/ a  —  x  —  ^  a 

_  A^ 

~      X 

[Several  of  these  equations  cau  be  noro  eb^aatly  reduced  by  the  method  given  on  p.  138» 
Ex.  47.] 


APPLICATIOKS  OF   SIMPLE  EQUATIONS. 


Applicatioks. 

29,  x^ccording  to  the  definition  (2),  Algebra  treats  of,  1st,  The 
nature  and  properties  of  the  Equation  ;  and  2d,  the  method  of  using 
it  as  an  instrument  for  mathematical  mvestigation. 

Having  on  the  preceding  pages  explained  the  nature  and  proper- 
ties of  tlie  equation,  we  now  give  a  few  examples  to  illustrate  its 
utility  as  an  instrument  for  mathematical  investigation. 

30,  The  Alf/ebraic  Solution  of  a  problem  consists  of  two 
parts : 

1st.  Tlie  Statement^  which  consists  in  expressing  by  one  or 
more  equations  the  conditions  of  the  problem. 

2d.  The  Solution  of  these  equations  so  as  to  find  the  values  of 
the  unknown  quantities  in  known  ones.  This  process  has  been 
explained,  in  the  case  of  Simple  Equations,  in  the  preceding  articles. 

31,  The  Slateinent  of  a  problem  requires  some  knowledge  of  the 
subject  about  which  the  question  is  asked.  Often  it  requires  a  great 
deal  of  this  kind  of  knowledge  in  order  to  "state  a  problem."  This 
is  not  Algebra ;  but  it  is  knowledge  which  it  is  more  or  less  important 
to  Ixave  according  to  the  nature  of  the  subject. 

32,  IPirections  to  guide  the  student  in  the  Statement  of  Prob- 
lems : 

1st.  Study  the  meaning  of  the  problem,  so  that,  \fyou  had  the  answer  given, 
yon  could  prate  it,  noWcmg  csLTGinWy  just  what  operations  you  would  have  to 
perform  upon  the  answer  in  proving.  This  is  called.  Discovering  the  relations 
between  the  quantities  involved. 

2d,  Represent  the  unknown  (required)  quantities  (the  answer)  by  some  one  or 
more  of  tlie  final  letters  of  the  alphabet,  as  t,  y,  z,  or  w,  and  the  known  quan- 
tities by  the  other  letters,  or,  as  given  in  the  problem. 

3d,  Lastly,  by  combining  the  quantities  involved,  both  knoiun  and  vnknown^ 
according  to  the  conditions  given  in  the  problem  (as  you  would  to  prove  it,  if  the 
ahsAver  were  known)  express  these  relations  in  the  form  of  an  equation. 

mi.  Sen. — It  is  not  always  expedient  to  use  x  to  represent  the  number 
sought.  Tlie  solution  is  often  simplified  by  letting  x  be  taken  for  some 
number  from  which  the  one  sought  is  readily  found,  or  by  letting  2x,  Sa;,  or 
some  multiple  of  x  stand  for  the  unknown  quantity.  The  latter  expedient 
is  often  used  to  avoid  fractions. 


90  ELEMENTARY   ALGEBRA. 


Problems. 

1.  A's  age  is  double  B's,  B's  is  triple  C's,  and  the  sum  of  their 
ages  is  140.     Required  the  age  of  eucli. 

2.  A's  age  is  m  times  B's,  B's  is  n  limes  C's,  and  the  sum  of  their 
ages  is  s.     liequired  the  age  of  each. 

3.  The  sum  of  two  numbers  is  48,  and  their  difference  12.  What 
are  the  numbers? 

4.  The  sum  of  two  numbers  is  «,  and  their  difference  d.  What 
are  the  numbers  ? 

5.  Having  the  sum  and  difference  of  two  numbers  given,  how  do 
you  find  the  numbers,  arithmetically  ? 

G.  A  post  is  -Jth  in  the  earth,  ^ths  in  the  water,  and  13  feet  in  the 
air.    What  is  the  length  of  the  post  ? 

7.  A  post  is  -itli  in  the  earth,  ^ths  in  the  water,  and  a  feet  in  the 
air.     What  is  the  length  of  the  post  ? 

8.  What  fraction  is  that,  whose  numerator  is  less  by  3  than  its 
denominator;  and  if  3  be  taken  from  the  numerator,  the  value  of 
the  fraction  will  be  J? 

9.  G'wQ  i\\Q general  &o\\\i\on  of  the  last;  /.c,  the  solution  when 
the  numbers  are  all  represented  by  letters.  Then  substitute  the 
above  numbers  and  find  the  answer  to  that  spetial  problem. 

SuG. — Letting  the  numerator  be  a  less  than  the  denominator,  and  -^  be  the 

am  +  bn 

fraction  after  b  is  taken  from  the  numerator,  the  fraction  is t-it- 

an  +  bn 

10.  A  man  sold  a  horse  and  chaise  for  1200 ;  J  the  price  of  the 
horse  was  equal  to  \  the  price  of  the  chaise.  Required,  the  price  of 
each.  Chaise,  $120  ;  horse,  $80. 

Generalize  and  solve  the  last,  and  then  by  substituting  the  numbers  given  in 
it  find  the  special  answers.     Treat  in  like  manner  the  next  nine  problems. 

11.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third  part,  21 
gallons  were  afterward  drawn,  when  it  was  found  that  one- half  re- 
mained.    How  much  did  the  cask  hold  ?  J «.s.,  126  galls. 

12.  A  and  B  can  do  a  piece  of  work  in  12  days,  but  when  A  worked 
alone  he  did  the  same  work  in  20.  How  long  would  it  take  B  to  do 
the  same  work  ?  Ans.^  30  days. 


APPLICATIONS  OF   SIMPLE  EQUATIONS.  91 

13.  A  cistern  ctui  be  filled  by  3  pipes;  by  the  first  in  IJ  hours,  by 
the  second  in  2J  hours,  and  by  the  third  in  5  hours.  In  what  time 
will  the  cistern  be  filled,  when  all  are  left  open  at  once  ? 

14.  Four  merchants  entered  into  a  speculation,  for  which  they 
subscribed  4755  dollars;  of  which  B  paid  tiiree  times  as  much  as  A; 
C  paid  as  much  as  A  and  B ;  and  D  paid  as  much  as  C  and  B.  What 
did  each  pay  ? 

15.  A  and  B  trade  with  equal  stocks.  In  the  first  year  A  tripled 
his  stock  and  had  $'27  to  spare ;  B  doubled  his  stock,  and  had  1153 
to  spare.  Now  the  amount  of  both  their  gains  was  five  times  the 
stock  of  either.     What  was  that  ? 

16.  A  and  B  began  to  trade  with  equal  snms  of  money.  In  the 
first  year  A  gained  40  dollars,  and  B  lost  40;  but  in  the  second  A 
lost  one-third  of  what  he  tiien  had,  and  B  gained  a  sum  less  by  40 
dollars  than  twice  the  sum  that  A  had  lost;  when  it  appeared  that 
B  had  twice  as  much  money  as  A.  What  money  did  each  begin 
with  ?  Ans.,  320  dollars. 

17.  What  number  is  that  to  which  if  1,  5,  and  13  be  severally 
added,  the  first  sum  divided  by  the  second  shall  equal  the  second 
divided  by  the  third? 

18.  Divide  49  into  two  such  parts  that  the  greater  increased  by  6 
divided  by  the  less  diminished  by  11,  shall  be  4|^. 

10.  A  cistern  which  contains  2400  gallons  can  be  filled  in  15 
minutes  by  three  pipes,  the  first  of  which  lets  in  10  gallons  per 
minute,  and  the  second  4  gallons  less  than  the  third.  How  much 
passes  througli  each  pipe  in  a  minute? 

20.  Find  a  number  such  that,  if  from  the  quotient  of  the  number 
increased  by  5,  divided  by  the  number  increased  by  1,  we  subtract 
the  quotient  of  3  diminished  by  the  number,  divided  by  the  number 
diminished  by  2,  the  remainder  shall  be  2. 

21.  Divide  a  into  two  such  parts,  that  one  may  be  the  ^th  part  of 
the  other. 

22.  Divide  a  into  two  such  parts,  that  the  sum  of  the  quotients 
which  are  obtained  by  dividing  one  part  by  m,  and  the  other  by  n, 

shall  be  equal  to  b.  The  parts  are    —^ ,  and  — . 

^  ^  n—m  7)1— n 


92  ELEMENTARY   ALGEBRA. 

J33.  Letting  p  represent  the  principal,  i  the  interest  for  time  7,  a 
the  amount,  and  r  ihe  per  cent,  for  a  unit  of  time,  produce  the  fol- 
lowing formnlcB,  and  give  their  meaning : 


(2.)  a=p^-i=p      ^^^ 


100' 

100  +  tr 


rp 


(5.)  p  = 


(6.)p 


tr  ' 
100^, 
100  +  tr 


24.  In  what  time  will  a  given  principal  double,  triple,  or  quadru- 
ple itself,  at  5^?  at  6^?  at  7^? 

25.  What  is  the  worth  of  a  note  of  1500  Nov.  2d,  1872,  which  is 
dated  Feb.  23d,  1870,  bears  12^  interest,  and  is  due  Jan.  1st,  1875, 
money  being  worth  7^  ?  A  ns.,  $087.23  + 

26.  On  a  sum  of  money  borrowed,  annual  interest  is  paid  at  5^. 
After  a  time  1200  are  paid  on  the  principal,  and  the  interest  on  the 
remainder  is  reduced  to  i^.  By  these  changes  the  annual  interest  is 
lessened  one-third.    What  was  the  sum  borrowed  ? 

27.  An  artesian  well  supplies  a  manufactory.  The  consumption 
of  water  goes  on  each  week-day  from  3  a.m.  to  6  p.m.  at  double  the 
rate  at  which  the  water  flows  into  the  well.  If  the  well  contained 
2250  gallons  of  water  when  the  consumption  began  on  Monday 
morning,  and  tlie  well  was  just  emptied  at  6  p.m.  on  the  next  Tliurs- 
day  evening  but  one,  how  many  gallons  flowed  into  the  well  per 
hour  ? 

28.  The  hind  and  fore  wheels  of  a  carriage  have  circumferences 
16  and  14  feet  respectively.  How  far  has  the  carriage  advanced 
when  the  fore  wheel  has  made  51  revolutions  more  tlian  the  other? 

29.  A  merchant  gains  the  first  year  15^  on  his  capital;  the  second 
year,  20^  on  the  capital  at  the  close  of  the  first;  and  the  third  year, 
25^  on  the  capital  at  the  close  of  the  second ;  when  he  finds  that  he 
has  cleared  $1000.50.     Required  his  capital.  Capital  |;1380. 

30.  A  man  had  12550  to  invest.  He  invested  part  in  certain  3^ 
stocks,  and  part  in  R.  R.  shares  of  $25  each,  wjiich  pay  annual  divi- 
dends of  $1.00  per  share.  The  stocks  cost  him  $81  on  a  hundred, 
and  the  R.  R.  shares  $24  per  share ;  and  his  income  from  each  source 
is  the  same.     Huw  many  R.  R.  shares  did  he  buy  ? 


SIMPLE  EQUATIONS   WITH  TWO   UNKNOWN   QUANTITIES.  93 


SECTION  II, 

INDEPENDENT,    SIMULTANEOUS,  SIMPLE   EQUATIONS   WITH   TWO 
UNKNOWN    QUANTITIES. 


Definitions. 

34.  Independent  Equations  are  such  as  express  different 
conditions,  and  neither  can  be  reduced  to  the  other. 

35.  Siinultiineoiis  JEquatlons  are  those  wliich  express  dif- 
ferent conditions  of  the  same  problem,  and  consequently  the  letters 
representing  the  unknown  quantities  signify  tlie  same  things  in  each. 
All  the  equations  of  such  a  group  are  satisfied  by  the  same  values  of 
the  unknown  quantities. 

36.  Eli^ninatlon  is  the  process  of  producing  from  a  given 
set  of  simulran^'ous  equations  containing  two  or  more  unknown 
quantities,  a  new  s^^t  of  equations  in  which  one,  at  lejist,  of  the  un- 
known quantities  shall  not  appear.  The  quantity  thus  disappearing 
is  said  to  be  eliminated.  (The  word  literally  inean.s  j^utting  out  of 
doors.     We  use  it  as  meaning  causing  to  disappear.) 

37.  There  are  Five  Methods  of  Elimination,  viz.,  by 
Cowpariso7i,  by  Substitution,  by  Addition  or  Subtraction,  by  Unde- 
termitied  Multipliers^  and  by  Division. 


Elimination  by  Comparison. 

38.  I^VOb.  1. — Having  given  two  inde]?endent,  simidtaiieous^ 
simple  equations  hetioeen  ttoo  tenknown  quantities,  to  deduce  therefrotn 
by  Comparison  a  new  equation  coiitaining  only  one  of  the  unknoicn 
quantities. 

RULE. — 1st.  Find  expressions  for  the  value  of  the  same 

UNKNOWN     QUANTITY    FROM     EACH    EQUATION,   IN    TERMS     OF    THE 
OTHER   UNKNOWN   QUANTITY   AND   KNOWN   QUANTITIES. 

2d.  Place  these  two  values  equal  to  each  other,  and  the 

RESULT  WILL   BE  THE   EQUATION   SOUGHT. 

Dem. — The  first  operations  being  performed  according  to  the  rules  for  simple 
equations  with  one  unknown  quantity,  need  no  further  demonstration. 


94  ELEMENTAIiY   ALGEBRA. 

2d.  Having  formed  expressions  for  the  value  of  the  same  unknown  quantity 
in  both  equations,  since  the  equations  arc  simultaneous  this  unknown  quantity 
means  the  same  thing  in  the  two  equations,  and  hence  the  two  expressions  for 
its  value  are  equal.     Q.  e.  d. 

ScH. — The  resulting  equation  can  be  solved  by  the  rules  already  given. 


Elimination  by  Substitution. 

39^  JProh,  2, — Having  given  two  independent^  simidtaneous, 
simple  equations,  between  two  unknown  quantities,  to  deduce  there- 
from by  Substitution  a  single  equation  with  btU  one  of  the  unknown 
quantities. 

RULE.— 1st.  Find  from  one  of  the  equations  the  value 
OF  the  unknown  quantity  to  be  eliminated,  in  terms  of  the 
other  unknown  quantity  and  known  quantities. 

2d.  Substitute  this  value  for  the  same  unknown  quan- 
tity IN  the  other  equation. 

Dem. — The  first  process  consists  in  the  solution  of  a  simple  equation,  and  is 
demonstrated  in  the  same  way. 

The  second  process  is  self-evident,  since,  the  equations  being  simultaneous, 
the  letters  mean  the  same  thing  in  both,  and  it  does  not  destroy  the  equality  of 
the  members  to  replace  any  quantity  by  its  equal,    q.  e.  d. 


Elimination  by  Addition  or  Subtraction. 

40,  I^vob,  3, — Having  given  two  independent^  simultaneous, 
simple  equations  betwee^i  two  unknown  quantities,  to  deduce  therefrom 
by  Additioji  or  Subtraction  a  single  equation  with  hut  one  unknown 
quantity, 

RULE. — 1st.  Reduce  the  equations  to  the  forms  ax  +  by 
=  m,  AND  ex  -[•  dy  =  n. 

2d.  If  the  coefficients  of  the  quantity  to  be  eliminated 
are  not  alike  in  both  equations,  make  them  so  by  finding 
their  L.  C.  M.  and  then  multiplying  each  equation  by  this 

L.    C.    M.     exclusive     of    the    factor    which    the    TERil    TO    BE 

eliminated  already  contains. 

3d.  If  the  signs  of  the  terms  containing  the  quantity  to 
be   eliminated  are  alike   in   both   equations,  subtract  onk 


SIMPLE   EQUATIONS — ELIMINATION.  95 

EQUATION   FROM   THE  OTHER,  MEMBER  BY  MEMBER.      If   THESE   SIGNS 
ARE   UNLIKE,   ADD   THE   EQUATIONS. 

Dem. — The  first  operations  are  perfQrmed  according  to  the  rules  already  given 
for  clearing  of  fractions,  transposition,  and  uniting  terms,  and  hence  do  not  viti- 
ate the  equations.  The  object  of  this  reduction  is  to  make  the  two  subsequent 
steps  practicable. 

The  second  step  does  not  vitiate  the  equations,  since  in  the  case  of  either 
equation,  both  its  members  are  multiplied  by  the  same  number. 

The  third  step  eliminates  the  unknown  quantity,  since,  as  the  terms  containing 
the  quantity  to  be  eliminated  have  the  same  numerical  value,  if  they  have  the 
saitie  sign,  by  subtracting  the  equations  one  will  destroy  the  other,  and  if  they 
have  different  signs,  by  adding  the  equations  they  will  destroy  each  other.  The 
result  is  a  true  equation,  since.  If  equals  (the  two  members  of  one  equation)  are 
added  to  equals  (the  two  members  of  the  other  equation),  the  sums  are  equal. 
Thus  we  have  a  new  equation  with  but  one  unknown  quantity,     q.  E.  D. 


Elimination  by  Undetermined  Multipliers. 

4:1.  I*VOb,  4, — Having  given  two  indepe7ident,  simultaneous, 
simple  equations  between,  two  unknown  quantities^  to  deduce  therefrom 
by  Undetermined  Midtipliers  a  single  equation  with  but  one  unknown 
quantity. 

RULE. — 1st.  Reduce  the  equations  to  the  forms  ax  -f  by 
=  m,  AND  ex  +  dy  =  n, 

2(1.    Multiply  one  of  the  equations  by  an  undetermined 

FACTOR,  AS  /,  and  FROM  THE  RESULT  SUBTRACT  THE  OTHER  EQUA- 
TION,   MEMBER   BY    MEMBER. 

3d.   In  the   resulting   equation,  place  the  coefficient  of 

THE  UNKNOWN  QUANTITY  TO  BE  ELIMINATED  EQUAL  TO  0;  FROM 
this  equation  find  the  value  of  /,  ANP  SUBSTITUTE  IT  IN  THE 
OTHER  TERMS   OF   THE   EQUATION. 

Dem. — [Reason  for  the  first  step,  same  as  in  the  last  method.] 

Now  multiply  one  of  the  equations,  as  «cj  +  by  =  m,  by  /,  and  subtract  tlio 
otuer,  member  by  member,  giving  (af—  c)x  +  (&/—  d)y  =  mf—  n.     To  eliminate 

y,  put  bf—d  =  0,  giving  f  =  -.       This  value  of  /  substituted  in  {af  —  c).v 

+  (V—  d)y  =  mf—  n,  will  cause  the  term  containing  y  to  disappear  by  making 
its  coefficient  0,  and  there  will  result  an  equation  containing  only  the  unknown 
quantity  x,  and  known  quantities.     Q.  E.  d. 


96  ELExMENTAKY   ALGEBRA. 

Thus,  given  3^  +  7^  —  33,  and  2.f  +  4y =20. 

Multiply  the  1st  by/, 3^iP  +  7/y  =  33/ 

Subtract  the  2d, 2x'  +   Ay  =  20 

And  we  have (3/- 2)*  +  (7/- %  =  33/- 20. 

Putting  7/  -  4  =  0,  /  =  f .     Substituting,  (3  x  ^-2)j!  =  33  x  *  -  20.      Whence, 

In  like   manner,  putting  3/  -  2  :=i  0,/=  f.     And  (7  x  ^  -  4)y  =  33  x  ^  -  20- 
Whence  y  =  3. 


Elimination  by  Division. 

42,  Proh,  S» — Havmg  yioen  t%co  independent^  sinndtaneoiis, 
equations  of  any  decp'ee^  betireen  two  tmknown  quaiitities^  to  deduce 
therefrom  by  Division  a  sinyle  equation  with  but  one  unknown 
quantity. 

RULE. — Clear  the  equations  of  fractions,  and  transpose 

ALL  the  terms  TO  ONE  MEMBER.  TrEAT  THE  POLYNOMIALS  THUS 
OBTAINED  AS  IN  THE  PROCESS  FOR  FINDING  THE  HIGHEST  COM- 
MON Divisor,  continuing  the    process   until   one    unknown 

QUANTITY  DISAPPEARS  FROM  THE  REMAINDER.  PUTTING  THIS  RE- 
MAINDER EQUAL  TO  0,  WE  HAVE  THE   EQUATION   SOUGHT. 

Dem. — Since  each  of  the  polynomials  is  equal  to  0,  any  number  of  times  one 
subtracted  from  the  other  (t.  c.  any  remainder)  is  0. 


Examples. 

[Note. — Tlie  pupil  should  solve  the  following  by  each  of  the  preceding 
methods,  so  as  to  make  all  familiar,  and  in  each  instance  notice  what  method  is 
most  expeditious.] 

(1.)    2x4-7j/=41,         (2.)      a:  +  152/  =  49,      (3.)  Qx+  4y=236, 
3a:  +  4?/=42.  3a;+   ltj=n,  3.r  +  15y=573. 

(4.)  %^x^\1b  =  Uy,     (5.)  188-5.r-9!/=0,  (6.)  bx-A=Zy, 

87a;-56y=497.  13a:=57  +  2y.  10  +  7a;- 12?/= 0. 

(7.)     5i/-21=  2rc,       (8.)    72/-3a:=139,       (9.)  ^^-\lx=   103, 
13a;-4?/=120.  2x-Vhy=  91.  Ux-l^=-4:l. 

(10.)     ?Z:?-l^-  =  ?^,  (11.)        ab.  +  ccly=% 

%y+i_ix+y  +  13  d-h 

-3- 1 -.  ax-cy=-^. 


SIMPLE   EQUATIONS — ELIMINATION.  97 

(12.)     T^=u-^,  (13.)  {b  +  c){x+c-b)+a(y-\-a)=2a^, 


b-\-y      '6a  +  x 
ax-\-%hy=d. 


ay       _  {h  +  cY 
{b-c)x  ~     fl2     • 


(15.)     2.4.+.32,-:?^^p!^=.8.  +  ?:^±^,    My^^^n^^ 

"2x      by      ^x      y 

i^a^     "^~i^       T~3       ^  ,     x-y       1 

(16.)     — --T —  =  2,    and     — -^  =  -. 

4  T 

(17.)     i  +  |-  =  5,     (18.)^+-=19,     (19.)  -^+^=m   +71, 
^     '     ax    by  ^     '  X      y  ^     '  nx      my  ' 

5       3       „  8      3^  w       ?;i  „       . 

^=2.  =7.  -  +  -    =m2+7z«. 

ax     by  ^^      y  ^       y 

[Note. — Solve  the  following  by  {42).] 

20.  Eliminate  x  between  the  following:  5x-{-y=106  and  x  +  Sy 
=35  ;  also  i(2a:  +  3y)  =  8-^:1'  and  U+y=l(7y-dx) ;  also  iiy-2) 
-i{^0-tj)={(z-10)  and  i(2z  +  A)-i{2y  +  z)  =  i(y+13);  also 
x^+6xy=U4:  and  6a-?/  +  36y2=432  ;  also  a:3  4-?/3=:2728  and  x^-xy 
+^2  — 124;  iilso  x^+x^yi-x^y^ -\-xy^ -\-y^  =  la,jid.x^ -{-y^=2;  also 
x-\-y  +  xy=M  and  a;2+?/2  =  52. 

SOLUTION   OP  THE  LAST. 

x-{-y  +  xy-34:  )  a;^ +i/g-52|a:4-34-y 

(l+y)a:  +  y-34  )  (l+y).r2  +  (l+i/)^2_52(i+y) 
(1  -\-y)x^  -\-xy—'d4:X 

(34-7/)a;+(:?/«-52)(l  +  i/) 
(l+y)(U-y)x  +  {y^'-o2)(l+yY 
(l+y){U-y)x-m-yy 
Equation  sought,  (34-?/)2 +  (;//2-52)(l  +  ?^)2=0. 

[Note. — The  equations  resulting  from  the  elimination  in  several  of  the  above 
cases  are  of  degrees  higher  than  the  first,  and  hence  their  reeolutipn  is  not  to 
be  expected  at  this  stage  of  the  student's  progress.] 

7 


98  ELEMENTARY   ALGEBRA. 


Applications. 

1.  A  wine  merchant  has  two  kinds  of  wine,  one  worth  72  cents  a 
quart,  and  the  other  40  cents.  How.  much  of  each  must  he  put  in  a 
mixture  of  50  quarts,  so  that  it  shall  be  worth  60  cents  a  quart? 

2.  A  crew  that  can  pull  at  the  rate  of  12  miles  an  hour  down  the 
stream,  finds  that  it  takes  twice  as  long  to  row  a  given  distance  up 
stream  as  down.     What  is  the  rate  of  the  current  ? 

3.  A  man  sculls  a  certain  distance  down  a  stream  which  runs  at  a 
rate  of  4  miles  an  hour,  in  1  hour  and  40  minutes.  In  returning  it 
takes  him  4  hours  and  15  minutes  to  reach  a  point  3  miles  below  his 
starting  place.  How  far  did  he  scull  down  the  stream,  and  at  what 
rate  could  he  scull  in  still  water? 

4.  A  man  puts  out  $10,000  in  two  investments.  For  the  first  he 
gets  5^  and  fur  the  second  4j^.  The  first  yields  annually  $50  more 
than  the  second.     What  is  each  investment? 

[Note. — Generalize  the  statement  and  solution  of  the  preceding  problems.] 

5.  AVhat  fraction  is  that  whose  numerator  being  doubled  and  de- 
nominator increased  by  7,  the  value  becomes  f;  but  the  denomina- 
tor being  doubled,  and  the  numerator  increased  by  2,  the  value  be- 
comes I  ? 

6.  There  is  a  number  consisting  of  two  digits,  which  is  equal  to 
four  times  the  sum  of  those  digits ;  and  if  18  be  added  to  it,  the 
digits  will  be  inverted.    What  is  the  number? 

7.  A  work  is  to  be  printed,  so  that  each  page  may  contain  a  cer- 
tain number  of  lines,  and  each  line  a  certain  number  of  letters.  If 
Ave  wished  each  page  to  contain  3  lines  more,  and  each  line  4  letters 
more,  then  there  would  be  224  letters  more  in  each  page;  but  if  we 
wished  to  have  2  lines  less  in  a  page,  and  3  letters  less  in  each  line, 
then  each  page  would  contain  145  letters  less.  How  many  lines  are 
there  in  each  page  ?  and  how  many  letters  in  each  line  ? 

8.  A  sum  of  money  put  out  at  simple  interest  amounted  to  $5250 
in  10  months,  and  to  $5450  in  18  months.  What  was  the  principal, 
and  what  the  rate  ? 


SIMPLE   EQUATIONS  WITH   TWO   UNKNOWN   QUANTITIES.  99 

9.  In  an  alloy  of  silver  and  copper,  —  of  the  whole  +  i?  ounces 

was  silver^  and  —  of  the  whole  —  q  onnces  was  copper.     How  many 
ounces  were  there  of  each  ? 

10.  When  a  is  added  to  the  greater  of  two  numbers,  it  is  m  times 
the  less;  but  when  h  is  added  to  the  less,  it  is  n  times  the  greater. 
What  are  the  numbers  ? 

11.  When  4  is  added  to  the  greater  of  two  numbers,  it  is  3;^  times 
the  less  ;  but  when  8  is  added  to  the  less,  it  is  J  the  greater.  What 
are  the  numbers?  Solve  by  substituting  in  the  results  of  the  pre- 
ceding. 

12.  There  is  a  cistern  into  which  water  is  admitted  by  three  cocks, 
two  of  which  are  of  exactly  the  same  dimensions.  When  they  are 
all  open,  five-twelfths  of  the  cistern  is  filled  in  4  hours ;  and  if  one 
of  the  equal  cocks  be  stopped,  seven-ninths  of  the  cistern  is  filled  in 
10  hours  and  40  minutes.  In  how  many  hours  woifld  each  cock  fill 
the  cistern  ? 

13.  A  banker  has  two  kinds  of  change ;  there  must  be  a  pieces  of 
the  first  to  make  a  crown,  and  h  pieces  of  the  second  to  make  the 
same  :  now  a  person  wishes  to  have  c  pieces  for  a  crown.  How  many 
pieces  of  each  kind  must  the  banker  give  him  ? 

Ans.,    —, of  the  first  kind,  -\ of  the  second. 

o—a  h—a 

14.  An  ingot  of  metal  which  weighs  7i  pounds  loses  jy  pounds  when 
weighed  in  water.  This  ingot  is  itself  composed  of  two  other  metals, 
which  we  may  call  M  and  M' ;  now  n  pounds  of  M  loses  q  pounds 
when  weighed  in  water,  and  7i  pounds  of  M'  loses  r  pounds  when 
weighed  in  water.  How  much  of  each  metal  does  the  original  ingot 
contain  ? 

Ans.,   — ^  pounds  of  M,  -^ — ^  pounds  of  M'. 

r  —  q     ^  r  —  q     ^ 


100  ELEMENTARY  ALGEBRA. 


A 


SECTION  III. 

INDEPENDENT,  SIMULTANEOUS,  SIMPLE  EQUATIONS  WITH  MORE 
THAN  TWO  UNKNOWN  QUANTITIES. 

43,  Prob, — Having  given  several  independent^  simultaneous, 
simple  equations^  involving  as  many  unknown  quantities  as  there  are 
equations,  to  find  the  values  of  the  unknown  quantities. 

RULE. — Combine  the  equations  two  and  two  by  any  of 

THE  METHODS  OF  ELIMINATION,  ELIMINATING  BY  EACH  COMBINA- 
TION THE  SAME  UNKNOWN  QUANTITY,  THUS  PRODUCING  A  NEW  SET 
OF  EQUATIONS,  ONE  LESS  IN  NUMBER,  AND  CONTAINING  AT  LEAST 
ONE  LESS  UNKNOWN  QUANTITY.  COMBINE  THIS  NEW  SET  TWO  AND 
TWO  IN  LIKE  MANNER,  ELIMINATING  ANOTHER  OF  THE  UNKNOWN 
QUANTITIES.  REPEAT  THE  PROCESS  UNTIL  A  SINGLE  EQUATION  IS 
FOUND  WITH  BUT  ONE  UNKNOW^N  QUANTITY.  SOLVE  THIS  EQUATION 
AND  THEN  SUBSTITUTE  THE  VALUE  OF  THIS  UNKNOWN  QUANTITY  IN 
ONE  OF  THE  NEXT  PRECEDING  SET  OF  EQUATIONS,  OF  WHICH  THERE 
WILL  BE  BUT  TWO,  WITH  TWO  UNKNOWN  QUANTITIES,  AND  THERE 
WILL    RESULT    AN    EQUATION    CONTAINING    ONLY    ONE,   AND    THAT 

another  of  the  unknown  quantities,  the  value  of  which 
can  therefore  be  found  from  it.  substitute  the  two  values 
now  found  in  one  of  the  next  preceding  set,  and  find  the 
value  of  the  remaining  unknown  quantity  in  this  equation. 
Continue  this  process  till  all  the  unknown  quantities  are 
determined. 

Dem. — 1.  The  combinations  of  the  equations  give  true  equations  because  they 
are  all  made  upon  the  methods  of  elimination  already  demonstrated. 

2.  That  the  number  of  equations  can  always  be  reduced  to  one  by  this  pro. 
cess,  is  evident,  since,  if  we  have  n  equations  and  combine  any  one  of  them  with 
each  of  the  others,  there  will  be  ?i.  —  1  new  equations.  Combining  one  of  these 
71  —  1  new  equations  with  all  the  rest  there  wiU  result  n  —  2.  Hence  n—\ 
such  combinations  will  produce  a  single  equation;  and  as  one  unknown  quan- 
tity, at  least,  has  disappeared  from  each  set,  there  will  be  but  one  left.     q.  e.  d. 

ScH.  1. — If  any  equation  of  any  set  does  not  contain  the  quantity  we  are 
seeking  to  eliminate,  this  equation  can  be  Avritten  at  once  in  the  next  set, 
and  the  remaining  equations  combined. 

ScH.  2. — ^In  eliminating  any  unknown  quantity  from  a  particular  set  of 


SIMPLE   EQUATIONS   WITH    SEVERAL   UNKNOWN    QUA^TJTJ!ES.     101« 

equations,  any  one  of  the  equations  may  be  combined  with  each  of  the 
others,  and  the  new  set  thus  formed.  But  some  other  order  may  be  prefer- 
able as  giving  simpler  results. 

ScH.  3. — It  is  sometimes  better  to  find  the  values  of  all  the  unknown 
quantities  in  the  same  way  as  the  first  is  found,  rather  than  by  substitution. 


Examples. 
1.  2.  3. 

x  +  y  +  z  =  31,       X+    y  +    z  =  9,  2x  +  dy  -h  4:Z  =.  29, 

X  +  y  —  z  =  25,        X  -i-  3y  —dz=  7,  dx  +  2y  +  5z  =  33, 

x  —  y  —  z=    9.        ic  —  4y  +  83  =  8.  4^;  +  3y  +  2^  =  25. 


5.  6. 

\x  +  \y  +  iz=Q2,     a;+i^=100,  >!    a:=64;      ?^±?^  +  2^=8,  1    a:=3; 

-^  I 


^^+iy  +  -i^=47,     ;y  +  l^  =  100, 
i-^+iy+i^=38.      z-\-\x=\0^. 


y=72',       x  +  2y-oz=2,  >  y=2; 


7. 
y  +  z 


2 

X  -{-  z 


x-h- 

y-     3 

,  +  ^  =  85. 


85, 
85, 


8. 

ay  +  bx  =  c, 

ex  +  az  =  b, 
hz  +cy  =  a. 


2      1_3 

X      y~  ^ 


y 


i  +  i  =  *. 

X        z        3 


10.  11. 

y-{-z=2yz, 

x-{-z=3xZf 

x  +  y=4:xy.  ^4-V=l.  bcx -\- cay  +  abz=l. 

13.     xyz=a{yz—zx—xy)=b{zx—xy—yz)=c(xy—yz—zx). 


12. 

a    b     ^ 

-+-=1,  o;+y  +  z=0, 

b     c 

-  +  -=1,         (b-hc)x-\-(c-}-a)y  +  {a-{-b)z=0, 

y    ^ 

Z       X 


'ttfjt^'ti      i  •*A  >  ELEMENTABY   ALGEBRA. 

14                                 15.  16. 

5x—'7z=n,              dx—5y  +  2z— 4:21  =  11,  n-\-v-{-x  +  z  =  ll, 

2a;  +  3y=39,            lOy-Sz -\-3u-2v=i  2,  2c  +  v  +  i/-\-z=12, 

4y+32=41.               5z-{-4:U-\-2v—2x=  3,  u-{-x-\-y  +  z=13, 

6ic—3v+4:X—27/=  6.  i; +.T +  ?/  +  ;?=:  14. 

17.    x+y  +  z=a-\-b  +  c,    hx-\-cy  +  az=cx^-ay  +  hz=a^ +J)* -\-c^. 


Applications. 

1.  Three  persons,  A,  B,  and  C,  were  talking  of  their  guineas; 
says  A  to  B  and  C,  give  me  half  of  yours  and  I  shall  have  34;  says 
B  to  A  and  C,  give  me  a  third  part  of  yours  and  I  sliall  have  34; 
says  C  to  A  and  B,  give  me  a  fourtii  part  of  yours  and  I  shall  have 
34.     How  many  had  each  ?  Ans.,  A  10,  B  22,  C  26. 

2.  For  $8  I  can  buy  2  lbs.  of  tea,  10  lbs.  of  coffee,  and  20  lbs.  of 
sugar,  or  2  lbs.  of  tea,  5  lbs.  of  coffee,  and  30  lbs.  of  sugar,  or  3  lbs. 
of  tea,  5  lbs.  of  coffee,  and  10  lbs.  of  sugar.    What  are  the  prices  ? 

3.  A  person  possesses  a  certain  capital  which  is  invested  at  a  certain 
rate  per  cent.  A  second  person  has  £1000  more  capital  than  the 
first  and  invests  it  at  one  per  cent,  more;  tlius  his  income  exceeds 
that  of  the  first  person  by  £80.  A  third  person  has  £500  more 
capital  than  the  second,  and  invests  it  one  per  cent,  more  advan- 
tageously ;  and  thus  receives  £70  more  income.  Find  the  capital  of 
eacii  and  the  rate  of  investment. 

4.  Find  four  nnmbers,  such  that  the  first  with  half  the  rest,  the 
second  with  a  third  the  rest,  the  third  with  a  fourth  the  rest,  and 
the  fourth  with  a  fifth  of  the  rest  shall  each  be  equal  to  a, 

5.  A  number  is  represented  by  6  digits,  of  which  the  left-hand 
digit  is  1.  If  the  1  be  removed  to  units  place,  the  others  remaining 
in  the  same  order  as  before,  the  new  number  is  3  times  the  original 
number.     Find  the  number. 

6.  A  man  has  £22  14,<?.  in  crowns  (55.),  guineas  (21^.),  andmoidores 
(275.) ;  and  he  finds  that  if  he  had  as  many  guineas  as  crowns,  and 
as  many  crowns  as  guineas,  he  would  have  £36  G.9. ;  but  if  he  had 
as  many  crowns  as  moidores,  and  as  many  moidores  as  crowns,  he 
would  have  £45  lGcS\     How  many  of  each  has  he? 


SIMPLE   EQUATIONS   WITH   SEVERAL   UNKNOWN   QUANTITIES.     103 

7.  A  person  has  four  casks,  the  second  of  which  being  filled  ft-om 
the  first,  leaves  the  first  -f  full.  The  third  being  filled  from  the 
second,  leaves  it  J  full;  and  when  the  third  is  emptied  into  the 
fourth,  it  is  found  to  fill  only  j\  of  it.  But  the  first  will  fill  the  third 
and  fourth  and  have  fifteen  quarts  remaining.  How  many  quarts 
does  each  hold  ? 

8.  A,  B,  C,  and  D,  engage  to  do  a  certain  work.  A  and  B  can  do 
it  in  12  days,  A  and  D  in  15  days,  and  D  and  C  in  18  days.  B  and  C 
commence  the  work  together,  after  3  days  are  joined  by  A,  and  after 
4  days  more  by  D.  Then,  all  working  together,  they  finish  it  in 
2  days.  How  long  would  each  have  required  to  do  the  entire  work? 
Solve  with  one  unknown  quantity,  as  well  as  with  four. 

9.  A  person  sculling  in  a  thick  fog.  meets  one  tug  and  overtakes 
another  which  is  going  at  the  same  rate  as  the  former ;  show  that  if 
a  is  thj  greatest  distance  to  which  he  can  see,  and  b,  V  are  the  dis- 
tances that  he  sculls  between  the  times  of  his  first  seeing  and  of  hid 

2      11 
passing  the  tugs,    -  =  .  +  t-,- 


104  ELEMENTARY  ALGEBRA. 


CHAPTER  II. 

BATIO,   PBOPORTION,  AND   mOGBESSION. 


SECTION  I. 
RATIO. 

44:.  Ratio  is  the  relative  magnitude  of  one  quantity  as  com- 
pared with  another  of  the  same  kind,  and  is  expressed  by  the  quotient 
arising  from  dividing  the  first  by  the  second.*  Tlie  first  quantity 
named  is  called  tlie  Antecedent^  and  the  second  the  Consequent. 
Taken  together  they  are  called  the  Terms  of  the  ratio,  or  a  Covplet. 

43,  The  Sif/n  of  ratio  is  the  colon,  : ,  the  common  sign  of 
dNision,  -^,  or  the  fractional  form  of  indicating  division. 

The  last  form  is  coining  into  use  quite  generally,  and  is  to  be  preferred. 

46,  Cor. — A  ratio  being  merely  a  fraction,  or  an  unexecuted 
problem  in  Division,  of  which  the  antecedent  is  the  numerator,  or 
dividend,  and  the  consequent  the  denominator,  oi'  divisor,  any  changes 
made  upon  the  terms  of  a  ratio  produce  the  same  effect  upon  its  value, 
as  the  like  changes  do  upon  the  value  of  a  fraction,  when  made  upon 
its  corresponding  terms.     The  priiicipal  of  these  are, 

1st.  If  both  terms  are  mtdtiplied,  or  both  divided  by  the  same 
number,  the  value  of  the  ratio  is  not  changed. 

2d.  A  ratio  is  multiplied  by  multiplying  the  antecede^it,  or  by 
dividing  the  consequent. 

3d.  A  ratio  is  divided  by  dividing  the  antecedent,  or  by  multiply- 
ing the  consequent. 

*  There  is  a  common  notion  among  us  that  the  French  exprcs*  a  ratio  by  dividing  the  con- 
Fequent  by  the  antecedent,  uhilo  the  English  express  it  as  above.  Such  is  not  the  fact. 
French.  German,  and  English  writers  agree  ia  the  above  definition.  In  fact,  the  Germans  very 
generally  use  the  sign  :  instead  of   ^^  and  by  all,  the  two  signs  arc  used  vm  exact  equivalents. 


RATIO.  105 

47 •  A  Direct  Ratio  is  the  quotient  of  the  antecedent  divided 
by  the  consequent,  as  explained  above,  (44).  An  Indirect  or 
Reciprocal  llatio  is  the  quotient  of  the  consequent  divided  by 
the  antecedent,  i.  e.,  the  reciprocal  of  the  direct  ratio.  A  ratio  is 
always  written  as  a  direct  ratio. 

48,  A  ratio  of  Greater  Inequality  is  a  ratio  which  is 
^a-eater  than  unity,  as  4  :  3.  A  ratio  of  Less  Inequality  is  a 
ratio  which  is  less  than  unity,  as  3:4. 

49,  A  Compound  liatio  is  the  product  of  the  corresponding 
terms  of  several  simple  ratios.  Thus,  the  compound  ratio  a  :  b, 
c  :  d,  m  :  n,  is  acm  :  bdn.  This  term  corresponds  to  cotnpound  frac- 
tion. A  compound  ratio  is  the  same  in  effect  as  a  compound  fraction. 

50,  A  Duplicate  Ratio  is  tlie  ratio  of  the  squares,  a  tri- 
plicatCf  of  the  cicbesy  a  snhduplicate^  of  the  square  roots,  and 
a  siibtriplicate,  of  the  cube  roots  of  two  numbers.    Thus,  aJ^  :  b^, 

«3    ;    ^3^      y^fi     .    .y/^^      r^ij^       y^^^     .     ,^^ 


Examples. 
1.  What  is  the-ratio  of  8  to  4  ?     of  4  to  8  ?    of  J  to  f  ?    of  6a^m 
to3am?    of  x^-y^  tox-y?    ofJto|?    of  -  to  |  ?    of  ^'~^^ 

91  0  X  —  X 

.    a  +  b^ 

to- ? 

1  —  X 

2.  Write  the  inverse  ratio  in  each  case  in  the  last  paragraph. 

3.  Reduce  the  following  to  their  lowest  terms:  85  :  187,  a^  —  b^ 
:  a*  -  b*,  n{a  -  xy  :  Q(a^  -  x^). 

4.  Wliat  is  the  duplicate  ratio  of  3:5,  of  rt:Z>?  What  the  tripli- 
cate ?  What  the  subduplicate  of  25 :  IG  ?  of  3  :  7  ?  of  m :  ^i  ?  What 
the  subtriplicate  of  729  :  1728  ?  of  .r :  y  ? 

5.  AVhich  is  the  greater,  the  compound  ratio  of  |:f  and  5:4,  or 
the  inverse  triplicate  ratio  of  3:2? 

6.  Prove  that  a  ratio  of  greater  inequality  is  diminished  by  adding 
the  same  number  to  both  its  terms.  How  is  it  with  a  ratio  of  less 
inequality?     How  with  equality? 

7.  If  5  gold  coins  and  30  silver  ones  are  worth  as  much  as  10  gold 
coins  and  10  silver  ones,  what  is  the  ratio  of  their  values  ? 


106  ELEMENTARY  ALGEBRA. 

8.  Prove  that  a^  —  x^ia^-hx^y  a—x:a-\-x.    Is  x^ -\- y^  :x^ +y^ 
greater,  or  less,  than  x*  +  i/^ :  x  -\-  y? 

9.  Prove  that  4:a^—3a^x  —  -kax^  +  3x^  :  da^—  2a^x  —  3nx^  +  2x^ 
is  equal  to  4^  —  3x :  3a  —  2x. 

10.  Prove  that,  if  a:  be  to  ^  in  the  duplicate  ratio  of  a  to  J,  and  a 
to  b  in  the  subduplicate  ratio  o^  a  +  x  to  a  —  y,  then  will  2x :  a 
^x-y:y. 


SECTION  If. 

PROPORTION. 

51,  Proportion  is  an  equality  of  ratios,  the  terms  of  the  ratios 
being  -expressed.  The  equality  is  indicated  by  the  ordinary  sign  of 
equality,  =,  or  by  the  double  colon,  : : . 

ScH. — The  pupil  should  practice  writing  a  proportion  in  the  fonn     r  =  ~,, 

still  reading  it  "a  is  to  &  as  c  is  to  rf."     One  form  should  be  as  familiar  as 
the  other.    He  must  accustom  himself  to  the  thought  that  a  :b  ::  c  :  d  means 

.-  =  --  and  nothing  mare. 
0      a 

52,  The  Extremes  (outside  terms)  of  a  i)roportion  are  the 
first  and  fourtli  terms.  The  Means  (middle  terms)  are  the  second 
and  third  terms. 

53,  A  Mean  Proportional  between  two  quantities  is  a 
quantity  to  which  either  of  the  othtr  two  bears  the  same  ratio  that 
the  mean  does  to  the  other  of  the  two. 

o4,  A  Tfiird  Proportional  to  two  quantities  is  such  a 
quantity  that  the  first  is  to  the  second  as  the  second  is  to  this  third 
(proportional). 

SS.  A  proportion  is  taken  by  Inversion  when  the  terms  of 
each  ratio  are  written  in  inverse  order. 

SO,  A  projwrtion  is  taken  by  Alternation  when  the  means 
are  made  to  change  places,  or  the  extremes. 

S7.  A  proportion  is  taken  by  Cotnjyosition  when  the  sum  of 
the  terms  of  eacli  ratio  is  compared  with  eitlier  term  of  that  ratio, 
the  same  order  being  observed  in  both  ratios ;  or  when  the  sum  of 


PROPORTION.  107 

the  antecedents  and  tlie  sum  of  the  consequents  are  compared  with 
either  antecedent  and  its  consequent. 

S8.  If  the  difference  instead  of  the  siim  he  taken  in  the  last  defi- 
nition, the  proportion  is  taken  by  Division. 

SO.  Four  quantities  are  Inverseh/  or  Reciprocally  Proportional 
wlien  the  first  is  to  the  second  as  the  fourth  is  to  the  third,  or  as  the 
reciprocal  of  the  third  is  to  the  reciprocal  of  the  fourth. 

60.  A  Continued  Proportion  is  a  succession  of  equal 
ratios,  in  which  each  consequent  is  the  antecedent  of  the  next  ratio. 
Thus  \i a-.b :\  b'.c '.:  c. d '.:  d: e,  we  have  a  continued  proportion. 


(il»  Prop.  1. — In  any  proportion  the  product  of  the  extremes 
equals  the  product  of  the  means. 

Dem. — If  a:b  ::  c:d then  ad  =  be.    For  a:b  ::  c:d  is  the  same  as  -  =  -,  which 

b       d 
cleared  of  fractions  becomes  ad  =  be.    Q.  E,  d. 

62,  Cor.  1. —  The  square  of  a  mean  proportional  equals  the  pro- 
duct of  its  extremes,  and  hence  a  mean  proportional  itself  equals  the 
square  root  of  the  product  of  its  extremes. 

If  a:m  ::  m:d,\)y  the  proposition  m*  =  ad.  Whence  extracting  the  square 
root  of  both  members,  m  ~  Vad. 

63.  Cor.  2. — Either  extreme  of  a  proportion  equals  the  ^^roduct 
of  the  means  divided  by  the  other  extreme;  and,  in  like  manner, 
either  mean  equals  the  product  of  the  extremes  divided  by  the  other 
mean. 


64.  Prop.  2, — Ty  the  product  of  two  quantities  equals  the  pro- 
duct of  two  others,  the  two  former  may  he  made  the  extremes,  or  the 
means  of  a  p7'oportion,  and  the  tioo  latter  the  other  terms. 

Dem. — Suppose  my  =  nx.  Dividing  both  members  by  xy,  we  have  —  =  -, 
i.  e.,m'.x  ::  n:y.     In  like  manner  dividing  by  w;i  we  have  -  =  — ,    i.  e.,  y.n 

11         TTh 


Deduce  each  of  the  following  forms  from  the  relation  my  =  nx 

1.  m  :  X  : ;  ??.  :  y. 

2.  m  :  n  ::  X  :  y. 

3.  y  :  n  •.:  X  :  m. 

4.  X   :  y  ::  m:  n. 


5. 

y 

X  : 

:  n 

m 

6. 

X 

m: 

•  y 

n. 

7. 

n 

m : 

••  y 

X. 

8. 

n 

y- 

:  m 

X. 

108  ELEMENTARY  ALGEBRA. 

6S»  Cor. — If  four  qumitities  are  in  proportion^  they  are  in  pro- 
portion  by  alternation  and  by  inversion. 


(jQ,  Prop,  3, — If  four  qua7itities  are  i?i  proportion,  the  propor- 
tion is  not  destroyed  by  taking  equal  multiples  of 

1st.   77/e  ie7')ns  of  the  same  couplet, 

2d.  7%e  antecedents, 

3d.  The  co7isequents, 

4th.  All  the  tei-ms. 

Demonstrate  these  facts  from  the  nature  of  a  proportion  as  an  equality  of 
ratios. 

07»  Sen. — Observe  that  such  changes,  and  only  such,  may  be  made  upon 
the  tenns  of  a  proportion  without  destroying  it,  jis 

1st.  Do  not  change  the  vdlues  of  the  ration, 

2d.    Change  loth  ratios  alike. 

Query. — If  the  first  term  of  a  proportion  be  divided  by  any  number,  in  what 
ways  may  the  operation  be  compensated  for  so  as  to  preserve  the  proportion  ? 


08,  JProj),  4, —  The  products  or  the  quotients  of  t/ie  corrtspond- 
ing  terms  of  two  {or  more)  proi)ortions  are  proportional  to  each 
other. 

Demonstrated  on  the  axioms  that  equals  multiplied  by  equals  give  equal 
products,  and  that  equals  divided  by  equals  give  equal  quotients. 

09.  Cor. — Like  powers,  or  roots,  of  proportionals  are  propor- 
tional to  each  other. 

How  does  this  corollary  grow  out  of  the  proposition  ? 


70,  Prop.  S. — If  two  proportions  have  a  ratio  in  one  equal  to 
a  ratio  in  the  other,  the  remaining  7'atlos  are  equal  and  may  form  a 
p>ro2yo7'tion. 

Demonstrated  on  the  axiom  that  things  which  are  equal  to  the  same  thing 
are  equal  to  each  other. 


Kl.  Pvop.  0, — Afiy  proportion  may  be  takeii  by  composition, 
or  by  division,  or  by  both  at  once,  without  destroying  it. 


PROPORTION. 

Dem.— If    a:b'.:c.d, 

a  +  h  :  h  :  :  c  +  d  :  d, 

(1) 

We  may   write   by 

a  +  b  :  a  :  :  c  +  d  :  c, 

(3) 

composition. 

a  +  c  :  a  :  :  h  +  d  :  b, 

(3) 

.  a  +  c  :  c  :  :  b  +  d  :  d. 

(4) 

10^ 


By  division,  we  may  write  the  same  forms  with  the  —  sign  instead  of  the  + . 

By  composition  and  f  a  +  b  :  a  -  b  :  :  c  +  d  :  c  -  d, 

division   at    the   same  < 

.,  \  a  +  c  :  a  —  c  :  :  b  +  d  :  b  —  d. 

time,  we  may  write,  v. 

These  forms  may  all  be  verified  by  representing  the  ratio  of  «  to  6  by  r, 

a 
whence   -r  =  r,  or  a  =  I'b,  and  since  the  ratio  of  c  to  d   is  the  same  as  that  of 

a  to  b,  -  =  r,  orc=  dr,  and  then  substituting  in  each  of  the  above  forms  these 
d 

values  of  a  and  c.  Thus,  the  Ist  becomes  br  +  b  :  b  ::  di'  +  d  :  d,  which  ratios  are 
equal,  smce  each  is  r  -i- 1 .  Let  the  student  verify  the  other  forms  in  the  same 
way. 

Queries. — If  a  :  b  :  :  c  :  d,\s  a±b  :  a  ::  c±d  :bt  Is  a-\-b  :  c  +  d  : :  a—c  :  b—dt 

72,  Cor. — If  there  be  a  series  of  equal  ratios  in  the  form  of  a 
conti I med  proportion^  the  sum  of  all  the  antecedents  is  to  the  sum  of 
all  the  consequents,  as  any  one  antecedent  is  to  its  consequent, 

Dexi. — If  a  :b  :  :  c  :d  :  :  e  :f:  :  g  :  h,  etc.,  a  +  c  +  e  +  g  +  etc. :  b  {-d  +/-I-  h  +  etc, 
:  :  a  :  &,  or  c  :  d,or  e  :f,OT  g  :  h,  etc.  Substitute  for  a  br,  for  c  dr,  for  e  fr,  for 
g  hr,  and  we  have 

br  +  dr+fr  +  hr  + etc.  :  b  +  d+f-\-7i  + etc.  :  :  h'  :  b, 
in  which  the  ratios  are  seen  to  be  equal,  since  each  is  r. 

73,  ScH. — TJie  method  pursued  in  tlie  demonstration  of  the  preceding  propo- 
sition wUl  be  found  sufficient  in  itself  to  test  any  projwsed  transformation  of  a 
propoHion.     We  will  give  a  few  examples  : 

1.  l{  a  :  h  :  :  c  :  d,  prove  as  tibove  that  ad  =  be. 

SUG. — By  substituting  as  above  we  have  the  identity  brd  =  bdr. 

2.  l^  (I  :  b  : :  c  :  d,  prove  as  above  that  a  :c  ::  b  :  d,  and  b  :  a::  d  :  c. 

3.  If  a  :  b  ::  c  :  d,  and  m  :  n  ::  x  :  i/,  prove  as  above  that  am  :  bn  : : 
ex :  dy. 

Sug's, — Let  —  =  r,  whence  —  =  r;  and  —  =  r' ,  whence  _  =  /.      Substitut- 
b  d  n  y 

ing  for  a  br,  for  c  dr,  for  m  nr' ,  and  for  x  yr' ,  in  the  proportion  to  be  tested,  it 

ie  shown  to  be  true. 


110  ELEMENTARY  ALGEBRA. 

4.  If  ^a  —  X  :  ^a  +  X  : :  b  —  1/  :  i)  -\-  J,  show  that  2x  :  y  '.:  a  :  b. 

Sug's. — From! =  r  find  x  in  terms  of  a  and  r,  and  from  . — ^  =  r  find  « 

ia  +  a;  &  +  y 

in  terms  of  &  and  r. 

5.  Ua:b::p:  q,  prove  that  a^  _|_  ^2  .  _fL_^  ..2^2  +  ^72  ;  _P , 

fl^  +  ^^  p  -^  q 

6.  Four  given  numbers  are  represented  by  a,  b,Cyd;  what  quantity 
added  to  each  will  make  them  proportionals? 

.  be  —  ad 

A71S., -_ 

a  —  0  —  c  -h  d 

7.  If  four  numbers  are  proportionals,  show  that  there  is  no  num- 
ber which,  being  added  to  each,  will  leave  the  resulting  four  num- 
bers proportionals. 

8.  If  a  :  b  ::  c  :  dy  show  that  via  :mb  ::  c:d;  a  :b  ::  mc :  md',  ma  : 
b  ::  mo  :  d;    a  :  mb  ::  a  :  md ;    and  ma  :71b::  mc  :  nd. 


Applications. 

[Note. — The  first  five  of  the  following  examples  should  be  solved  without 
converting  the  proportions  into  equations.] 

1.  A  merchant  having  mixed  a  certain  number  of  gallons  of  brandy 
and  water,  found  that  if  he  had  mixed  6  gallons  more  of  each,  there 
would  have  been  7  gallons  of  brandy  to  every  6  gallons  of  water, 
but,  if  he  had  mixed  6  gallons  less  of  each,  there  would  have  been 
0  gallons  of  brandy  to  every  6  gallons  of  water.  How  much  of  each 
did  he  mix  ? 

Solution,     a;  +  6  :  y  +  6  : :  7  :  6,  and  a;  -  6  :  y  -  6  : :  6  :  5. 
Ilencc  «  -  y  :  y  +  6  : :  1  :  6,  and  a;  —  y  :  y  —  6  : :  1  :  5. 

Hence  y  +  6  :  y  -  6  : :  6  :  5,  or  2y  :  12  : :  11  :  1,  or  y  :  66  : :  1  : 1. 

Substituting,     «  +  6  :  72  : :  7  :  6,  or  «  +  6  :  6  : :  14  :  1,  or  .t :  6  : :  13  :  1,  or  a? : 
1  : :  78  :  1. 

2.  Find  two  numbers,  such,  that  their  sum,  difference,  and  pro- 
duct, may  be  as  the  u umbers  5,  d,  and;j,  respectively. 

Solution,     x  +  y  :  x  —  p  ::  s  :  d,  and  x  —  y:xy::d:p. 
Hence  <i^  ■  p ':  s  -i-  d  :  s  -  d,  and  x  :  p  ::  dx  +  p  :  p. 

Hence  dx+p  :p::  s  +  d:  8-  d,ordx  \  p  ::  2d  :a  -  d,OT  X  :  p  ::  2  :«-(!, 

OTXil  ■..2p:8-d,i.e.x  =  —^. 
8  —  d 


PROPORTION.  HI 

3.  It  is  required  to  find  a  number,  such,  that  the  sum  of  its  digits 
is  to  the  number  itself  as  4  to  13 ;  and  if  the  digits  be  inverted,  their 
difference  will  be  to  the  number  expressed  as  2  to  31. 

4.  Divide  the  number  14  into  two  such  parts,  that  the  quotient 
of  the  greater  divided  by  the  less  shall  be  to  the  quotient  of  the  less 
divided  by  the  greater,  as  16  to  9. 

5.  Find  two  numbers  whose  difference  is  to  the  difference  of  their 
squares  as  m  :  n,  and  wTiose  sum  is  to  the  difference  of  their  squares 
RS  a  :b. 

[Note. — In  tlie  following,  use  the  proportion  more  or  less,  as  is  found  ex- 
pedient.] 

6.  The  sides  of  a  triangle  are  as  3  :  4  :  5,  and  the  perimeter  is  480 
yards:  find  the  sides. 

7.  A  fox  makes  4  leaps  while  a  hound  makes  3 ;  but  two  of  the 
hound's  leaps  are  equivalent  to  3  of  the  fox's.  What  are  their  relative 
rates  of  running? 

8.  A  courier  sets  out  from  Trenton  for  Washington,  and  travels 
at  the  rate  of  8  miles  an  hour;  two  hours  after  his  departure 
another  courier  sets  out  after  him  from  New  York,  supposed  to  be 
68  miles  distant  from  Trenton,  and  travels  at  the  rate  of  12  miles  an 
hour.  How  far  must  the  second  courier  travel  before  he  overtakes 
the  first  ? 

9.  There  are  two  places,  154  miles  apart,  from  which  two  persons 
set  out  at  the  same  time  to  meet,  one  travelling  at  the  rate  of  3  miles 
in  two  hours,  and  the  other  at  the  rate  of  5  miles  in  four  hours.  How 
long,  and  how  far,  did  each  travel  before  they  met  ? 

10.  A  courier,  who  travels  60  miles  a  day,  has  been  dispatched 
five  days,  when  a  second  is  sent  to  overtake  him,  in  order  to  do 
which  he  must  travel  75  miles  a  day.  In  what  time  will  he  overtake 
the  former? 

11.  Two  travellers,  A  and  B,  set  out  at  the  same  time  from  two 
different  places,  C  and  D;  A  from  C  to  D,  and  B  from  D  to  0. 
When  they  met,  it  appeared  that  A  had  gone  30  miles  more  than 
B ;  also,  that  A  can  reach  D  in  4  days,  and  B  can  reach  C  in  9  days. 
Required  the  distance  from  C  to  D. 

12.  A  hare,  50  of  her  leaps  before  a  greyhound,  takes  4  leaps  to 
the  greyhound's  3;  but  two  of  the  greyhound's  leaps  are  as  much  as 


112 


ELEMENTARY  ALGEBnA. 


three  of  the  hare's.     How  many  leaps  must  the  greyhound  take  to 
catch  the  hare  ? 

13.  A  runuer  left  this  place  n  days  ago,  at  the  rate  of  a  miles 
daily.  He  is  pursued  by  another,  at  the  rate  of  b  miles  a  day.  In 
how  many  days  will  the  second  overtake  the  first  ? 

.  mi 

Ans.,  -r . 

0  —  a 

14.  Find  the  time  between  3  and  4  when  the  hands  of  a  watch  are 
opposite  each  other.  When  they  are  at  right  angles  to  each  other. 
When  they  are  together. 

15.  How  often  does  the  minute  hand  of  a  watch  pass  the  hour 
hand  ?    How  often  at  right  angles  ?    How  often  opposite? 

16.  Do  the  hands  of  a  watch  occupy  the  three  relative  positions 
of  opposite,  at  right  angles,  and  together  between  each  two  hours  of 
the  12  ?  If  there  are  exceptions  point  them  out,  and  show  why  they 
occur. 

17.  Before  noon,  a  clock  which  is  too  fast,  and  points  to  afternoon 
time,  is  put  back  5  hours  and  40  minutes ;  and  it  is  observed  that 
the  time  before  shown  is  to  the  true  time  as  29  to  105.  Required 
the  true  time. 

18.  Two  bodies  move  uniformly  around  the 
circumference  of  the  same  circle,  which  measures 
8  feet.  When  they  start,  one  is  a  feet  before  the 
other;  but  the  first  moves  m  and  the  s^^cond  M 
feet  in  a  second.  AVhen  will  these  bodies  pass 
each  other  the  first  time,  when  the  second,  when 
the  third,  etc.,  supposing  that  tliey  do  not  disturb 
each  other's  motion  ?  When  will  they  pass  if 
the  first  starts  t  seconds  before  the  second,  and  M  >  m?  When  if 
M  <  m  ?  When  will  they  pass  if  the  first  starts  /  seconds  later  than 
the  second  and  M  >  m  ?  When  if  M  <  m  ?  When  will  they  meet 
if  tliey  start  at  the  same  time  and  move  toward  each  other,  or  over 
tlie  distance  a,  first  ?  If  they  move  from  each  other,  or  over  the  arc 
s  —  a  first  ?  When  will  they  meet  if  ihi^  first  starts  /  seconds  before 
the  other,  and  they  move  toward  each  other,  or  over  the  distance  a 
first?  If  they  move  from  each  other,  or  over  the  arc  s  —  a  first? 
If  they  move  in  opposite  directions,  and  the  first  starts  t  seconds 
later  than  the  second?  When  they  move  over  the  arc  a  first?  When 
they  move  over  the  arc  s  —  a  first  ? 


PROGRESSIONS.  113 

19.  The  force  of  gravitation  is  inversely  as  the  square  of  the  dis- 
tance from  the  centre  of  the  earth.  At  the  distance  1  from  the 
centre  of  the  earth  this  force  is  expressed  by  the  number  32.16.  By 
vrhat  is  it  expressed  at  the  distance  60  ?  Ans.,  0.0089. 

20.  If  the  velocity  of  one  body  moving  around  another  is  propor- 
tional to  unity  divided  by  the  duplicate  of  the  distance,  and  the 
velocity  be  represented  by  v  when  the  distance  is  r,  by  what  will  it 
be  expressed  when  the  distance  is  r'  ? 

Ans.,  -Tiv 


SECTION  III, 

PROGRESSIONS. 


74,  A  JProgvession  is  a  series  of  terms  which  increase  or  de- 
crease by  a  common  difference,  or  by  a  common  multiplier.  The 
former  is  called  an  Arithmetical,  and  the  latter  a  Geometrical  Pro- 
gression. A  Progression  is  Increasing  or  Decreasing  according  as 
the  terms  increase  or  decrease  in  passing  to  the  right.  The  terms 
Ascending  and  Descending  are  used  in  the  same  sense  as  increasing 
and  decreasing,  respectively.  In  an  Arithmetical  Progression  the 
common  difference  is  added  to  any  one  term  to  produce  the  next  term 
to  the  right.  If  the  progression  is  decreasing  the  common  difference 
is  mimis.  In  an  increasing  Geometrical  Progression  the  constant 
multiplier  by  which  each  succeeding  term  to  the  riglit  is  produced 
from  the  preceding  is  more  than  unity;  and  in  a  decreasing  progres- 
sion it  is  less  than  unity.  This  constant  multiplier  in  a  Geometrical 
Progression  is  called  the  Ratio  of  the  series. 

t5.  The  character,  ••,  is  used  to  separate  the  terms  of  an  Arith- 
metical Progression,  and  the  colon, : ,  for  a  like  purpose  in  a  Geo- 
metrical Progression. 

ILLUSTRATIONS. 

1  ••  3  ••  5  ••  7,  etc.,  etc.,  is  an  increasing  Arithmetical  Progression  with  a  common 

difference  2,  or  +  3. 
15-10-5  -O"  —  5,  etc.,  etc.,  is  a  Decreasing  Aritnmetical  Progression  with  a 
common  difference  —  5. 
a  '•  a  ±  d  ••  a  ±  2d  ■•  a  ±  M,  etc.,  etc.,  is  the  general   form  of  an   Arithmetical 
Progression,  ±  d  being  the  common  difference. 

2  :  4  :  8  :  16,  etc.,  etc.,  is  an  increasing  Geometrical  Progression  with  ratio  2. 
12  :  4  :  ^  :  ^  :  jV,  etc.,  etc.,  is  a  Decreasing  Geometrical  Progression  with  ratio  i 

8 


114  ELEMENTARY   ALGEBRA. 

ai  ar  :  ar^  :  ar^  :  ar^,  etc.,  etc.,  is  the  general  form  of  a  Geometrical  Progres- 
sion, r  being  the  ratio,  and  greater  or  less  than  unity, 
according  as  the  series  is  increasing  or  decreasing. 

76.  When  three  quantities  taken  in  order  are  in  arithmetical  pro- 
gression, the  second  is  t^e  Arithmetical  Mean  between  the  other  two, 
and  is  equal  to  half  their  sum. 

III. — If  « ••  &  ••  c,  6  is  the  arithmetical  mean  between  a  and  c ;  and  since  h  —  a 
=  c  —  b,  b  =  \{a  +  c). 

77,  When  three  quantities  taken  in  order  are  in  geometrical  pro- 
gression, the  second  is  the  Geometric  Mean  between  the  other  two, 
and  is  equal  to  the  square  root  of  their  product. 

Let  the  student  illustrate. 


78.  There  are  Five  Tilings  to  be  considered  in  any  progression ; 
Tiz.,  the  first  term,  the  last  term,  the  common  difference  or  the  ratio, 
the  number  of  terms,  and  the  sum  of  the  series,  any  three  of  whicli 
being  given  the  other  two  can  be  found,  as  will  appear  from  the  sub- 
sequent discussion. 


Arithmetical  Progression. 

79,  I^rop.  1, —  The  formula  for  finding  the  nth,  or  last  term  of 
an  Arithmetical  Progressio7i ;  or,  more  properly,  the  formula  express- 
ing the  relation  between  the  first  term,  the  nth  term,  the  common  dif- 
ference, and  the  7iumber  of  terms  of  such  a  series,  is 

1  =  a  +  (n  -  l)d, 

in  which  a  is  the  first  term,  d  the  common  difference,  n  the  number 
of  terms,  and  1  the  nth  or  last  term,  d  being  positive  or  negative 
according  as  the  seHes  is  increasing  or  decreasing. 

Dem. — According  to  the  notation,  the  series  is 

a  '■  a  +  d  ■'  a  +  2d  ••  a  +  dd   -  a  -!-  Ad  ••  a  +  M,  etc.,  etc. 
Hence  we  observe  that  as  each  succeeding  term  is  produced  by  adding  the  com- 
mon difference  to  the  preceding,  when  we  have  reached  the  nth  term,  we  shall 
have  added  the  common  difference  to  the  first  term  n—1  times  ;  that  is, the  nth 
term,  or  I  =  a  +  {n  —  \)d.     q.  e.  d. 

Sen. — As  this  formula  is  a  simple  equation  in  terms  of  a,  I,  n,  and  <Z,  any 
ouc  of  them  may  be  found  in  terms  of  the  other  three. 


ARITHMETICAL  PROGRESSION.  llo 

SO,  Pvop.  2, —  7%6  formulcL  for  the  sum  of  an  Arithmetical 
Progression^  or  expressing  the  relation  between  the  sum  of  the  series^ 
the  first  term^  last  tei'ni,  and  number  of  terms,  is 

s  representing  the  sum  of  the  series^  a  the  first  term^  1  the  last  term^ 
ami  11  the  number  of  terms. 

Dem. — If  I  is  the  last  term  of  the  progression,  the  term  before  it  is  I  —d,  and 
the  one  before  that  ^— 2rf, etc.  Hence,  as  a-' a  +  d-'a  +  ^id'-a  +  'dd I  re- 
presents tlie  series,  l-l—d-l—2d--l—Zd a  represents  the  same  series 

reversed.     Now  the  sum  of  the  first  series  is 

»=o4-(«  +  <Z)  +  (a  +  2<^+ il—M)  +  {l—d)-k-l'y 

and  reversed         8=1  ■{■{l —d)-\-{l —'ii,d)-\-  -  -  -  {a  +  2d)  +  {a  +  d)-{-a. 

Adding  28={a  +  l)  +  {a  +  l)  +{a-\-l)->r  -  -  {a  +  l)  +  {a  +  l)-^{a  +  l). 

If  the  number  of  terras  in  the  series  is  n,  there  will  be  n  terms  in  this  sum,  each 

of  which  is  ('fc  +  ^) ;  hence  2«  =(«4-0''.  or  «  =    "o"    "•     ^'  ^'  ^* 

ScH. — This  formula  being  a  simple  equation  in  terms  of  «,  a,  Z,  and  n, 
any  one  of  the  four  can  be  found  in  tenns  of  the  other  three. 


51,  Cor.  1. — Formulas 

(1)  l=a  +  (n-l)d,        and 

(2)  s  =1^*— ~Jn,  being  two  equations  between 

the  five  quantities,  a,  1,  n,  d,  and  s,  any  two  of  these  five  can  be  found 
in  terms  of  the  other  three. 

52,  Cor.  2. — The  formula  for  inserting  a  given  number  of  arith- 
metical means  between  two  given  extremes  is  d=: -,  in  which  m 

represents  the  number  of  means.     Froyn  this  d,  the  common  differ- 
ence, being  found,  the  terms  can  readily  be  written. 

Dem. — If  a  is  the  first  term  and  I  the  last,  and  there  are  m  terms  between, 
or  m  means,  there  are  in  all  m.  +  3  terms.     Hence,  substituting  in  the  formula 

I  — a 
l—a-\-Oi-l)d,ior  ;?,  m  +  2,  wc  have  l-.-a  +  {in+'l)d.    From  this  <Z="— ^.    Q.  e.  d. 


116 


ELEMENTARY  ALGEBRA. 


S3.  FoRMULiE  IN  Arithmetical  Progression. 

[It  will  afford  a  good  exercise  for  the  student  to  solve  the  following  cases  on 
review,  after  having  gone  through  Quadratics ;  though  no  importance  need  be 
attached  to  remembering  the  results,  as  the  fundamental  formulas 

(1)  l=a  +  {7i-l)d,    and    (2)  «=r^Jw, 
are  sufficient  to  resolve  all  cases.] 


NUMBEU.      !                    GIVEN. 

I 

BEQUIRSD. 

FORMULAS. 

1. 

2. 
3. 

4. 

a,     d,     n 
a,     d,     S 

a,     n,     S 
d,     n,     S 

I 

.     l=a  +  {n-\)d. 

n 
n          2 

5. 
6. 

7. 
8. 

a,     d,     n 
a,     d,     I 

a,     n,     I 
d,     n,     I 

s 

S=in{2a  +  (w-l>?}, 
^-   2    +     2d    ' 

S=\n{2l-{n'-l)d\. 

9. 
10. 
11. 
12. 

d,      n,     I 
d,     n,     S 
d,     I,      S 
n,     I,      S 

a 

a=l-{n-\)d, 
^_S_{n-l)d 
n        2      ' 
a-^d±^{l+id)^-2d^, 

a= — —L 
n 

13. 
14. 
15. 
16. 

a,     n,     I 
a,     n,     S 
a,     I,      S 
n,     I,      S 

d 

l-a 

2(S-aw). 
^-  n{n-\) 

2(nl-S) 

17. 
18. 
19. 

20. 

a,     d,     I 
a,     d,     S 
a,     I,      S 
d,     I,      S 

n 

l-a     ^ 

±  \/{2a-dY  +  M^-2a  +  d 
""=                     2d 
2S 

2l  +  d±^{2l  +  dY-M^ 
^-                   2d 

GEOMETRICAL  PROGRESSION.  117 

Examples. 

1.  Find  the  21st  term  of  3  ••  7  ••  11  ••  etc.,  and  the  sum  of  these 
terms. 

2.  Find  the  24th  term  of  7--5--3--  etc.,  and  the  sum  of  these 
terms. 

3.  Find   the  nth  term  of  |..|..f..  etc.,  and  the  sum  of  the   n 
terms. 

4.  Find  the  ?ith  term  of  ^^—~  ••  -^^-  -^^•.  etc.,  and  the  sum 

n  n  n 

of  the  n  terms. 

5.  Insert  four  uritlimetical  means  between  193  and  443. 

6.  Prove  that  tlie  sum  of  7i  terms  of  1  ••  3  ••  5  ••  7  ••  etc.,  is  to  the 
sum  of  m  terms  as  n^   :  7)i^. 

.7.  What  is  the  first  term  of  an  arithmetical  progression  whose 
59th  term  is  -:l\,  aiidGOth  -If?     Whose  2d  term  is  i,  and  55th  5.8? 

8.  How   many  terms   in  the  progression  wliose  common  differ- 
ence is  3,  first  term  5,  and  last  term  302  ? 

9.  Insert  three  arithmetical  means  between  w?  and  n. 

10.  Produce  the  formula  for  inserting  m  arithmetical  means  be- 
tween a  and  h,  viz., 

am  +  h     am  — a -i- 2b                      bm  —  h-\-2a     bin  +  a     , 
d . . . . _----_ . . .  . .  ^. 

VI  + 1  m  +  l  VI  + 1  m  +  1 

11.  If  a  body  falling  to  the  earth  descends  a  feet  the  first  second, 
Sa  the  second,  5a  the  third,  and  so  on,  how  far  will  it  fall  during  the 
^th  second  ?  Ans.,  (2t  —  1  )a. 

12.  If  a  body  falling  to  the  earth  descends  a  feet  the  first  second, 
3a  the  second,  5a  the  third,  and  so  on,  how  far  will  it  fall  in  t 
seconds?  Aois-^at^. 


Geometrical  Progression. 

84,  JPrO]},  1, —  The  formula  for  finding  the  nth,  or  last  term 
of  a  f/eowetrical  prof/ressio7i ;  or,  more  properly,  the  formula  ex- 
pressinr/  the  relation  between  the  first  term^  the  wth  term,  the  ratio, 
and  the  number  of  terms  of  such  a.  series,  is  1  =:ar°~\  in  which,  1  is 
the  last^  or  nth  term,  a  the  first  term,  r  the  ratio ^  and  n  the  number  of 
term^. 


118  KLEMENTAKY   ALGEBRA. 

Dem, — Letting  a  represent  the  first  term  and  r  the  ratio,  the  series  is 
a  :  ar  :  «;•-  :  ar^  :  ar^  :  etc.  Whence  it  appears  that  any  term  consists  of  the 
first  term  multiplied  iuto  the  ratio  raised  to  a  power  whose  exponent  is  one 
less  than  the  number  of  the  term.    Therefore  the  ni\\  term,  or  I  =.  ar'*~  \    Q.  e.  d. 

83,  J^i'Op.  2 ,— The  formula  for  the  sum  of  a  geometrical  j^ro- 
gresslon,  or  exjjressvig  the  relation  between  the  sum  of  the  series,  the 
frst  tei')n,  the  ratio,  and  the  number  of  terms  is 

ar"  —  a 

^  =  T3T' 

in  which  s  represents  the  sum,  a  the  first  term,  r  the  ratio ^  and  n  the 
number  of  terms. 

Dem.— The  sum  of  the  series  being  found  by  adding  all  its  terms,  we  have, 
»  =  a -\-  ar  -\-  at*  ->r  ar^  -v  -  -  ar*-*  +  ar»-'  +  ar"-',  and  multiplying  by  r, 
r8  =        ar  +  ar*  +  ar'  +  . .  ar"-»  +  gr"-*  -f-  ar*-^  +  «?•*.    Subtracting, 
rs  —  s  =  ar*  —  a,    or 
{r  —  i)s  =  ar*  —  a,  and  s  — — -.    q.  e.  d. 

T  —  1 

^6*.  Cor.  1. — Formulas 

(1)  1  =  ar»-*,  and 

ar"  —"  '1 

(2)  s  =  — ; -^  being  two  equations  be- 
tween the  five  quantities,  a,  1,  r,  n,  and  s,  are  sufficient  to  determine 
any  two  of  them  when  the  others  are  given. 

87,  Cor.  2. — Since  1  =  ai-""*,  lr=ar",  which  substituted  in   (2) 

Ir  —  a 
gives  s  —  -; — -;  which  formula  is  often  co7ivenient. 

88,  Cor.  3. — The  formula  for  inserting  m  geometrical  means 

heticeen  a  and  1  is  x  =     y  -. 

^    a 

89,  Cor.  4. —  The  formula  for  the  sum  of  an  iiifinite  decreasing 
geometrical  progression  is  s  = . 

Dem. — Since  in  a  decreasing  progression  the  ratio  is  less  than  unity,  the  last 
term,  ar"-^,  is  also  less  than  the  first  term,  and  numerator  and  denominator  of 

the  value  of  s, ,  become  negative.    Hence  it  is  well  enough  to  write  the 

formula  for  the  sum  of  such  a  series  s  =  zr— — ,  that  is,  change  the   signs  of 

1  —  r 

both  terms  of  the  fraction.  Now,  if  the  terms  of  a  series  are  constantly  decreas- 
ing, and  the  number  of  terms  is  infinite,  we  can  fix  no  value,  however  small, 
which  will  not  be  greater  tlmn  the  last,  or  than  some  term  which  may  be 
reached  and  passed.     Hence  we  are  compelled  to  call  the  last  term  of  such  a 

series  0,  which  makes  the  formula  s  — .     Q.  E,  D. 


GEOMETRICAL  PROGllESSION. 


119 


90,  Geometrical  Formula. 

[In  a  review,  after  the  pupil  has  been  through  the  hook,  it  will  be  a  good  exer- 
cise for  him  to  deduce  the  following  formulas  from  the  two  fundamental  ones?. 
It  is  not  necessary  to  memorize  these,] 


10. 

11. 
12. 


GIVEN. 


a,  r,  n 

a,  r,  S 

a,  11,  S 

r,  n,  S 


a,  r,     7i 

a,  r, 

a,  n, 

r,  n, 


r,  n, 

r,  n,  S 

r,  I,  S 

n,  I,  S 


REQUIRED. 


I  =  «?•»-', 

^      a  +  (r-l)S 

~"  r 

I  (s  -  1)n-^  -  «(S  -  a)"-'  =  0, 
^^(r-l)S7-'>-' 
r»  —  1 


S: 
S: 


a(r»  —  1) 
rl  —  a 


r-1* 

.-I   -        n-l 

V  ^"  —    V  *** 
^r"  — ; 


(r  -  1)S 
r»-l  ' 
rl  -  (?•  -  1)S, 


=  0. 


13. 
14. 
15. 
16. 


a,  n,  I 

a,  n,  S 

a,  I,  S 

71,  I.  S 


?•«■ 


— r  + =  0, 

a  a 

S-a 


S  ? 

s - r     ^  s-i 


17. 
18. 
19. 
20. 


a,  r,  I 

a,  r,  S 

a,  I,  S 

r,  I,  S 


log  r 
log  [g  +  (r  —  1)S]  —  log  a 
log  r 
log  I  —  log  a 


+  1, 


log  (S  -  a)  -  tog  (S  -  /) 
logl-log[lr-{r-\)S-]   ^   ^ 
log  r 


120  ELEMENTARY  ALGEBBA. 


Examples. 

1.  In  a  geometrical  progression  the  first  term  is  3,  the  ratio  5,  and 
the  number  of  terms  7.     What  is  the  last  term  ?    What  the  sum  ? 

2.  Insert  5  geometrical  means  between  2  and  1458. 

3.  Find  the  llth  term  of -^  :  ^^  :  ^  :  etc.,  and  the  sum  of  the 
11  terms. 

4.  Find  the  7th  term  of  -  | :  J  :  ~  J  :  etc.,  and  the  sum  of  the  7 
terms. 

5.  Insert  4  geometrical  means  between  ^  and  ^^. 

6.  Find  the  sum  of  3  :  J  :  ^  :  etc.,  to  infinity.  Also  of  ^i  —J 
•.etc.,  to  infinity.    Also  of  .54.     Also  of  .836. 

7.  If  a  body  move  20  miles  the  first  minute,  19  miles  the  second, 
18^  the  third,  and  so  on  in  geometrical  progression  forever,  what  is 
the  utmost  distance  it  can  reach  ?  A?is.,  400  miles 

8.  What  is  the  distance  passed  through  by  a  ball,  before  it  comes 
to  rest,  which  falls  from  tiie  height  of  50  feet,  and  at  every  fall 
rebounds  half  the  distance,  the  time  of  ascent  equalling  the  time  cf 
descent?  Ans.,  150. 

9.  In  the  preceding  problem,  suppose  the  body  falls  16^  feet  the 
first  second,  3  times  as  far  the  next  second,  and  5  times  as  far  the 
third  second,  and  so  on,  how  long  will  it  be  before  it  comes  to  rest  ? 

Ans.,  3V^V'579(4  +  3\/2)  =  10.27657  +  seconds. 

10.  Find  the  sum  of  the  following  series : 

i—i  +  i—-^+  etc.,  to  n  terms. 

l+i  +  i  +  ^+  etc.,  to  10  terms.    Also  to  infinity. 

l|^  +  .5+  etc.,  to  12  terms.    Also  to  infinity. 

11.  To  find  what  each  payment  must  be  in  order  to  discharge  a 
given  principal  and  interest  in  a  given  number  of  equal  payments  at 
equal  in  tervals  of  time. 

Solution. — Let  p  represent  the  principal,  r  tlie  rate  per  cent.,  t  one  of  the 
equal  intervals  of  time,  n  tlie  number  of  payments  (i.  e.,  nt  is  the  whole  time), 
and  X  one  of  the  payments. 

There  will  be  as  many  solutions  as  there  are  different  methods  of  computing 
interest  on  notes  upon  which  partial  payments  have  been  made. 

1st.   By  the   United  States  Court  Mule. — As  the  payments  must  exceed  the 


GEOMETRICAL  PROGRESSION.  121 

interest  in  order  to  discharge  the  principal,  this  rule  requires  that  we  find  the 

vt 
amount  of  p,  for  time  t,  at  r  per  cent.     This  is  done  by  multiplying  by  1  +  ~ — , 

100 

and  gives  |>(1  +  ■ — )•  From  this  subtracting  the  payment  r,  the  new  prin- 
cipal is  j?{  1  +  —  )  —X.  Again,  finding  the  amount  of  this  for  another  period 
of  time,  t,  and  subtracting  the  second  payment, 

^\     100/        V      100/ 

In  like  manner,  after  the  third  payment  there  remains 

After  the  4th  payment,  the  remainder  is 

^\        100/  \        100/  V         100/  \        100/ 

Finally,  after  the  ni\\  payment,  we  have 

•^V        100/  V        100/  \        100/  V        100/ 

-x(l+lL\-x  =  0. 
\        100/ 

Whence 


This  denominator  being  the  sum  of  a  geometrical  progression  whose  first  term 

(\  +  ^V 

/         rt\  ^        100/  -  1 

is  1,  ratio  ( 1  +  — ),  and  number  of  terms  n,  its  sura  is   - 
V        100/ 

100  \     ^  100/ 


100 


Hence  x  = 


(-^r- 


2d.  By  the  Vermont  Rule. — The  amount  of  the  principal  for  the  whole  time 


122  ELEMENTAKY  ALGEBRA. 

The  amount  of  the  1st  payment  is xVl  -f- _-(7j,  —  1)1 

"        Sd  "  x[l+Jl{n-2)], 

L         100  J 

"3d  «  ........    x\l+  Ii(n-d)], 

L        100^  U' 

etc.,  etc.,  ........  etc. 

The  ?ith  payment  (with  no  interest)  is x. 

The  sum  of  the  amounts  of  these  payments  is 

''"'  +  ]^*[('^-  1)  +  0^  -  2)  +  (^  -  3) 1]. 

The  series  in  the  brackets  being  an  arithmetical  progression  whose  first  term 
is  {n  —  1),  common  difference  —  1,  last  term  1,  and  number  of  terms  {n  —  1),  its 

)n.    Hence  the  sum  of  the  payments  ie  tix  +  ^^x  (^  ~     iw.. 

2    /  ^  ^  100     \    2     / 


sum  IS 

nrt 


(n-l)- 


r     100^    n 

or  a;  L^  H o -•  •    ^^*  ^^  *^®  condition  this  sum  equals  the  amount  of 

the  principal ;  consequently 

ScH. — If  the  payments  are  made  annually,  t  =  l.     And  letting  r'=  — , 

i.  e.,  letting  the  rate  per  cent,  be  expressed  decimally,  the  formulas  become. 

By  the  U.S.  Rule,  ^^?>r'(l  +  rr 

(1  +r')"— 1 

By  the  Vermont  Rule,       x  =    ^^(^  +/'^)  . 

2n  +  rn{n—t) 

12.  Whafc  must  be  the  annual  payment  in  order  to  discharge  a  note 
of  $5000,  bearing  interest  at  10^  per  annum,  in  5  equal  payments  ? 
Ans.,  By  the  U.  S.  Rule,  $1318.99  within  a  half  cent. 
By  the  Vermont  Rule,  $1250. 

Query. — What  occasions  the  great  disparity  between  the  payments  required 
by  the  different  rules  ! 


VARIATION.  123 


SECTION  IV. 

VARIATION. 

91,  Variation  is  a  term  applied  to  the  consideration  of  quan- 
tities so  related  to  each  other  that  any  change  in  one  makes  the 
others  change  in  the  same  ratio,  4ii*ect  or  inverse. 

One  quantity  varies  directly  as  another,  when  any  change  in  the 
latter  makes  the  former  change  in  the  same  {direct)  ratio. 

One  quantity  varies  inversely  as  another,  when  any  change  in  the 
latter  makes  the  former  change  in  the  corresponding  inverse  ri^tio. 

Ill's.— The  amount  earned  by  a  laborer  in  a  given  time  varies  directly  as  his 
daily  wages.  The  time  required  to  earn  a  given  amount  varies  inversely  as  the 
daily  wages. 

92,  One  quantity  wsines  joiiitly  as  two  others,  when  any  change  in 
the  product  of  the  latter  two  makes  the  former  change  in  the  same 
ratio  as  this  product. 

III. — The  amount  a  laborer  receives  varies  jointly  as  his  daily  wages  and  the 
time  of  service. 

93,  One  quantity  varies  directly  as  a  second  and  inversely  as  a 
third,  when  it  varies  as  the  quotient  of  the  second  divided  by  the 
third. 

III. — The  time  required  to  earn  any  amount  varies  directly  as  the  amount, 
and  inversely  as  the  daily  wages. 

94,  The  Sign  of  variation  is  a. 

III. — If  X  varies  directly  as  y,  we  write  x  cc  y,  and  read  "  x  varies  as  y."  If  x 
varies  inversely  as  y,  we  write  a?  a  - ,  and  read  "  x  varies  inversely  as  y"    If  x 

y 

Y&Ties  jointly  as  y  and  z,  we  write  x  a  yz,  and  read  "  x  varies  jointly  as  y  and  z." 
If  X  varies  directly  as  y  and  inversely  as  z,  we  write  a;  a  — ,  and  read  **  x  varies 
directly  as  y,  and  inversely  as  z." 

95^  Prop, — Variation  may  always  be  expressed  in  the  form  of  a 
proportion. 

Dbm. — 1st.  The  expression  x  ex.  y  signifies  that  if  x  is  doubled  y  is  double  1, 
if  X  is  divided  y  is  divided  by  the  same  number,  etc. ;  i.  e.,  that  the  ratio  of  x  to 

y  is  constant.     Let  m  be  this  ratio,  so  that  —  =  m.    Therefore  x  :  y  : :  m  :  1. 


124  ELEMENTAllY  ALGEBRA. 

2d.  The  expression  a;  a  —  signifies  that  if  y  is  multiplied  by  any  number,  x  is 

divided  by  the  same,  and  if  y  is  divided  by  any  number  x  is  multiplied  by  the 
same.  Hence  the  product  of  x  and  y  is  constant.  Let  this  product  be  in.  Then 
xy  =  m,ov  x:l  ::  m:y. 

3d.    X  (X  yz  signifies  that  the  ratio  of  x  to  yz  is  constant.    Let  this  be  m.    Then 
—  =  m,  OT  x:yz  ::  mil,  or  x :  y  : :  mz  :  1,  or  x  :  z  : :  my  :l,or  x  :y  ::  2  :  — . 

yz  Wi 

4th.    X  cx^  signifies  that  the  ratio  of  x  to  -^  is  constant.  Let  this  be  m.  Then 

z  z 

x:  ^::m:l,  or  x:y::m:z 

z  . 


Exercises. 

1.  If  a:  a  y.  and  y  (x  z,  show  that  x  a  z. 

Dem. — If  X  (X  y,  the  ratio  of  a;  to  y  is  constant.  Let  this  ratio  be  m.  Then 
X  =  my.  In  like  manner  let  n  be  the  ratio  of  y  to  z.  Then  y  =  nz.  Hence 
X  =  mnz.    That  is,  the  ratio  of  .r  to  z  is  constant,  or  x  a  z. 

2.  If  a;  a  -,  and  y  a  -,  show  that  x  (x  z. 

y  ^      z 

Sug'8.— We  may  write  a;  =  — ,  and  y  — —.  Hence  x-=''^z.  That  is,  the  ratio 
of  a;  to  2  is  constant,  or  a;  a  z. 

3.  If  o;  a  z,  and  ?/  a  -,  show  that  x  <x  -. 

'         -^       z  y 

SuG's.    x  =  mz,  y  =  —,  .•.x  =  —,otxcc—. 

z  y  y 

X        v 

4.  If  a;  a  y,  show  that  -  a  -^,  and  xz  a  yz. 

X       V 

5.  If  a;  a  y,  and  z  a  n,  show  that  xz  ex.  yu,  and  -  oc  -. 

6.  If  a;  a  y,  and  y*  <x  z*,  how  does  x  vary  in  respect  to  z  ? 

7.  If  a;  a  ;/,  and  for  x  =  8,  ?/  =  4,  what  is  the  value  of  y  for 
a;  =  20  ? 

Solution.— Since  x  ocy,  and  for  a:  =  8,  y  =  4,  the  ratio  of  a;  to  2^  is  2.  That 
is,  -  =  2.    Hence  for  x  =  20,  we  have  ??  =  2,  or  y  =  10. 

y  y 

8.  If  a:  a  -  and  for  x  =  G,  y  =^2,  what  is  the  value  of  xfory  =3? 

SuG.    x:—  : :  6 :  - .    Hence  for  y  =  3,  .t  =  4.     Or  we  may  reason  thus,  in 

changing  from  2  to  3,  y  increases  |  times.     Then,  as  x  changes  in  the  reciprocal 
ratio,  cc  =  §  of  6  =  4. 


HARMONIC     PROPORTTOX     AND     PROGRESSION.  125 

9.  If  a  -}-  b  <x  a  —  b,  prove  that  a^  -\-  b^cc  ab. 

10.  If  y  =  2^  -{-  q,  in  which  p  o:  x  and  qcc-;   and  if  when  x  =  1, 

4         14 
?/  =  G  ;  and  when  x  =  2,  y  =  5;  prove  that  y  ^  -x  -{-  —. 

11.  The  area  of  a  triangle  equals  half  the  product  of  the  base  and 
altitude.  Show  that  if  the  base  is  constant  the  area  varies  as  the 
altitude ;  if  the  altitude  is  constant  the  area  varies  as  the  base  ;  and 
if  the  area  is  constant  the  altitude  and  base  vary  inversely. 

12.  The  volume  of  a  pyramid  varies  jointly  as  its  base  and  alti- 
tude. A  pyramid  whose  base  is  9  feet  square,  and  height  10  feet, 
contains  10  cubic  yards.  What  must  be  the  height  of  a  pyramid 
with  a  base  3  feet  square  in  order  that  it  may  contain  2  cubic  yards? 

13.  Given  that  .^  a  t^,  when  /  is  constant;  and  s  a/,  when  t  is 
constant;  also,  2s  ==/,  when  ^  =  1.  Find  the  equation  between  /,  s, 
arid  t. 

SuG. — The  first  two  conditions  are  equivalent  to  saying  that  s  varies  jointly  as 
t^  and  /,  i.  e.  8  <x  ft'^  ;  since  in  this  expression  if  /  is  constant  s  a  t^,  and  if  t  is 
constant  «  a  /. 


SECTION  V, 

HARMONIC  PROPORTION  AND  PROGRESSION. 

OG,  Three  quantities  are  in  Harmonic  Proportion  when  the  dif- 
ference between  the  first  and  second  is  to  the  difference  between  the 
second  and  third  (the  differences  being  taken  in  the  same  order)  as 
the  first  is  to  the  third. 

III.  6,4,  and  3  are  in  harmonic  proportion,  since  6  —  4  :  4  —  3  : :  6  :  3.  \i  a,h, 
c  are  in  harmonic  proportion,  a  —  b  :  b  —  c  : :  a  :  c. 

07,  Def. — When  three  quantities  taken  in  order  are  in  har- 
jnonic  proportion,  the  second  is  the  Harmonic  Mean  between  the 
other  two. 

08,  JProp, — If  three  quantities  are  in  harmonic  proportion^  their 
reciprocals  are  in  arithmetical  proportion. 

Dem.— If  a,  b,  c  are  in  harmonic  proportion,  a  —  b  :  b  —  c  '.  :  a  :  c,  and 

ac  —  be  —  ab  —  ac.    Dividing  by  abc,  we  have   . = r .  i-  c-  —  .. . 

^     J        '  b       a       c        b  abc 


126  ELEMENTARY  ALGEBRA. 

09,  Def. — The  reciprocals  of  the  terms  of  an  arithmetical  pro- 
gression form  what  is  called  a  Harmonic  Progression, 

III.— Thus  as  1,  2,  3,  4,  5,  6  is  an  arithmetical  progression.  1,  — ,  — ,  — ,  — ,  — 

A     o    4    5     6 
is  a  harmonic  progression.    Also  if  a,  b,  c,  d,  etc.,  constitute  a  harmonic  progres- 
sion, — ,  -r,  — ,  -:,  etc.,  consfitute  an  a^rithmetical  progression. 
abed 

100.  ScH. — The  term  Harmonic  is  applied  to  such  a  series,  since,  if  strings 
of  the  same  size,  substance,  and  tension,  be  taken  of  the  lengths  1,  ^,  i,  i,  i,  ^ , 
any  two  of  them  vibrating  together  produce  harmony  of  sound. 


Exercises. 

1.  If  a,  h,  c,  d  are  in  harmonic  progression,  show  that  ab  :  ccl  :: 
a  —  h  :  c  —  d. 

SuG  8. -J-  ••  —  ••  -J.    Hence  -j- =  — ,or  acd  —  bed  =  abc  —  abd. 

abed  bade 

2.  If  a,  h,  e  are  in  harmonic  proportion,  show  that  J  (the  harmonic 
mean)  = . 

3.  Show  that  the  geometric  mean  between  two  numbers  is  a  geo- 
metric mean  between  their  arifchmetio  and  harmonic  means. 

4.  To  insert  n  harmonic  means  between  a  r«nd  h. 

SuG. — First  find  the  form  of  the  terms  for  n  arithmetical  m«ans  befn^een 
-1  and  i.   See  [82).    The  hftfmo^ic  series  i^  a    "^'^  "^  ^^       ^^^  "^  ^^ 


«  b  '  '     bn  +  a    '  bn  +  2a  —  b  ' 

db(n  +  1)  ^ 

r— ,  b. 

an  +  b 

5.  If  a  and  b  are  the  first  two  terms  of  a  harmonic  progression, 

show  that  the  7ith  term  is  — ,^    ,, -. 

a(n  —  1)  —  b(7i  —  2) 

6.  Insert  3  harmonic  means  between  \  and  ■^. 

ScH.— There  is  no  method  known  for  finding  the  sum  of  a  harmonic 
series. 


PURE   QUADRATICS.  127 


CHAPTER  III. 

QUADRATIC    EQUATIONS. 


SECTION  I, 
PURE   QUADRATICS. 

101,  A  Quadratic  Equation  is  an  equation  of  the  second 
degree  {6,  8). 

102,  Quadratic  Equations  are  distinguished  as  Pure  (called  also 
Incomplete),  and  Affected  (called  also  Complete), 

103,  A  Pure  Quadratic  Equation  is  an  equation  which 
contains  no  power  of  the  unknown  quantity  but  the  second;  as 
ax^  -\-h  =  cd,  x^  -  U  =  102. 

104,  An  Affected  Quadratic  Equation  is  an  equation 
which  contains  terms  of  the  second  degree  and  also  of  the  first,  with 
respect  to  the  unknown  quantity  or  quantities;  as  x^  —  4a:  =  12, 
bxy  —  X  —  y^  =  16a,  mxy  ■\-  y  =  h. 

105,  A  Root  of  an  equation  is  a  quantity  which  substituted  for 
the  unknown  quantity  satisfies  the  equation. 


103,  Prob. — To  solve  a  Pure  Quadratic  Equation. 

PULE. — Transpose  all  the  terms  containing  the  unknown 

QUANTITY  INTO  THE  FIRST  MEMBER,  AND  UNITE  THEM  INTO  ONE, 
CLEARING  OF  FRACTIONS  IF  NECESSARY.  TRANSPOSE  THE  KNOWN 
TERMS  INTO  THE  SECOND  MEMBER.  DiVIDE  BY  THE  COEFFICIENT  OF 
THE  UNKNOWN  QUANTITY.  FINALLY,  EXTRACT  THE  SQUARE  ROOT 
OF  BOTH   MEMBERS. 

Dem. — According  to  the  definition  of  a  Pure  Quadratic,  all  the  terms  contain- 
ing the  unknown  quantity  contain  its  square.  Hence  they  can  be  transposed 
And  united  into  one  by  adding  with  reference  to  the  square  of  the  unknown 


128  ELEMENTARY   ALGEBRA. 

quant  it}'.  Tliat  transposition,  and  division  of  both  members  by  the  same  quan- 
tity, do  not  destroy  the  equality  has  already  been  proved.  Extracting  the  square 
root  of  the  first  member  gives  the  first  power  of  the  unknown  quantity,  i.  e.  the 
quantity  itself.  And  taking  the  square  root  of  both  members  does  not  destroy 
the  equation,  since  like  roots  of  equal  quantities  are  equal. 

107,  COE.  1. — Ei:ery  Pure  Quadratic  Equation  has  two  roots 
^numericalhj  equal  but  with  ojyposite  signs. 

For  every  such  equation,  as  the  process  of  solution  shows,  can  be  reduced  to 
the  form  ar'  =  a  {a  representing  any  quantity  whatever).  Whence,  extracting 
the  root,  we  have  x  =  ±  V~a  ;  as  the  square  root  of  a  quantity  is  both  + ,  and 
-  {203,  Part  I). 

108.  Cor.  2. —  The  roots  of  a  Pure  Quadratic  Equation  may 
both  be  imaginary,  a^id  both  loill  be  if  one  is. 

For  if  after  having  transposed  and  reduced  to  the  form  x^  =  a,  the  second 
member  is  negative,  as  a;'  =  —  a,  extracting  the  square  root  gives  x  =  +  V—a, 
and  X  —  —  V—a,  both  imaginary. 


Examples. 


a      \/(i*--x*__x  .        45     _     57 
x'^        X        ~b'  '  2a;2+3~4.r2_5* 

5  1  1  _Va  ^    x^  -V>_x^-4: 

Va^x+V~a     Va-^x—Va      x  3  4 

7.  x^—ax-hb=ax{x—l).  8.  8-^'Sx^  =  6  +  2x^, 


'■  f1?^-|/?^=*- 


10.  *  +  ^         ' 


4  +  9?j~2-.t' 

11.  12  +  4(.'r2  +  12)  =  (2-a:)(2  +  a;)-16.      12.  xV6Tx^=l+x^. 

._    ax  +  l  +  Va^x^  —  l     ,_  _    a  +  x-{-\/2ax-j-x^     ^ 

13.  =:=ibx.  14.  — =b' 

ax-\-l  —  ya^x^  —  l  a-hx—V2ax  +  x^ 

Applications. 

1.  Fiiul  two  numbers  which  shall  be  to  each  other  as  3  to  5,  and 
the  ditiercncc  of  whose  squares  shall  be  25G. 


PURE   QUADRATICS.  1'20 

2.  Find  a  number  such  that  if  the  square  root  of  LIio  diiference 
between  the  square  of  the  number  and  a^,  be  successively  subtracted 
from  and  added  to  a,  the  difference  of  the  reciprocals  of  these  results 
shall  be  equal  to  a  divided  by  the  square  of  the  number. 

3.  Find  three  numbers  which  shall  be  to  each  other  as  m,  7i,  and 
p,  and  the  sum  of  whose  squares  shall  be  s. 

4.  An  army  was  drawn  up  with  5  more  men  in  file  than  in  rank, 
but  when  the  form  was  changed  so  that  there  were  845  more  in 
rank,  there  were  but  5  ranks.  How  many  men  were  there  in  the 
army? 

5.  From  two  towns,  m  miles  asunder,  two  persons,  A  and  B,  set 
out  at  the  same  time,  and  met  each  other,  after  travelling  as  many 
days  as  are  equal  to  the  difference  of  miles  they  travelled  per  day, 
when  it  appeared  that  A  had  travelled  n  miles.  How  many  miles 
did  each  travel  per  day  ? 

6.  For  comparatively  small  distances  above  tlie  earth's  surface  the 
distances  through  vviiich  bodies  fall  under  the  influence  of  gravity 
are  as  the  squares  of  the  times.  Thus,  if  one  body  is  falling  2 
seconds  and  another  3,  the  distances  fallen  through  are  as  4:9.  A 
l)ody  falls  4  times  as  far  in  2  seconds  as  in  1,  and  9  times  as  far  in  3 
seconds.  These  facts  are  learned  both  by  observation  and  theoret- 
ically. It  is  also  observed  that  a  body  falls  16^  feet  in  1  second. 
How  long  is  a  body  in  falling  500  feet  ?  One  mile  (5280  ft.)  ?  Five 
miles  ? 

7.  The  mass  of  the  earth  is  to  the  mass  of  the  sun  as  1 :  354930, 
and  attraction  varies  directly  as  the  mass  and  inversely  as  the  square 
of  the  distance.  The  distance  between  the  earth's  centre  and  eund 
centre  being  91,430,000  miles,  find  the  point  between  the  earth  and 
sun  where  the  attraction  of  the  earth  is  equal  to  that  of  the  sun.  The 
earth's  radius  being  3,962  miles,  where  is  this  point  situated  with 
reference  to  the  earth's  surface  ? 

8.  A  certain  sum  of  money  is  lent  at  5^  per  annum.  If  we  multiply 
the  number  of  dollars  in  the  principal  by  the  number  of  dollars  in 
the  interest  for  3  months,  the  product  is  720.    What  is  the  sum  lent  ? 

9.  The  intensity  of  two  lights,  A  and  B,  is  as  7  :  17,  and  their  dis- 
tance apart  132  feet.  Where  in  the  line  of  the  lights  are  the  points 
of  equal  illumination,  assuming  that  the  intensity  varies  inversely 
as  the  square  of  the  distance  ? 


130  ELElrtfil^fAltt    ALGfi^'TtA. 

10.  The  loudness  of  one  church  bell  is  three  times  that  of  another 
Now,  supposing  the  strength  of  sound  to  be  inyersely  as  the  square 
of  the  distance,  at  what  place  on  the  line  of  the  two  will  the  bells  be 
equally  well  heard,  the  distance  between  them  being  a  ? 


SECTION  IT. 
AFFECTED  QUADRATICS. 


109,  An  Affected  Quadratic  equation  is  an  equation  which 
contains  terms  of  the  second  degree  and  also  of  the  first  with  respect 

to  the  unknown  quantity,    x*  —ox=  12,  4.^  +  ^ax^  =  - 


5 


2^8 


a*x 


and  -j-^  —  4:ax  +  3J*  =  0  are  affected  quadmtic  equations. 
110,  Prob, — To  solve  an  Affected  Quadratic  Equation. 

RULE.— I.  l^EDUCE  THE  EQUATION  TO  THE  FORM  X^  -^  OX  =  h, 
THE  CHARACTERISTICS  OF  WHICH  ARE,  THAT  THE  FIRST  MEMBER  CON- 
SISTS OF  TWO  TERMS,  THE  FIRST  OF  WHICH  IS  POSITIVE  AND  SIMPLY 
THE  SQUARE  OF  THE  UNKNOWN  QUANTITY,  ITS  COEFFICIENT  BEING 
UNITY,  WHILE  THE  SECOND  HAS  THE  FIRST  POWER  OF  THE  UNKNOWN 
QUANTITY,  WITH  ANY  COEFFICIENT  (a)  POSITIVE  OR  NKGATIVE, 
INTEGRAL  OR  FRACTIONAL;  AND  THE  SECOND  MEMBER  CONSISTS  OF 
KNOWN  TERMS  {b). 

2.  Add  the  square  of  half  the  coefficient  of  the  second 
term  to  both  members  of  the  equation. 

3.  Extract  the  square  root  of  each  member,  thus  producing 

A    SIMPLE   equation   FROM   WHICH    THE   VALUE   OF  THE    UNKNO^V:?^^ 
QUANTITY  IS  FOUND  BY  SIMPLE  TRANSPOSITION. 

Dem. — By  definition  an  affected  quadratic  equation  contains  but  three  kinds 
of  terms,  viz :  terms  containing  the  pquare  of  the  unknown  quantity,  terms  con- 
taining the  first  power  of  the  unknown  quantity,  and  knoicn  terms.  Hence  each 
of  tht  three  kinds  of  tenns  may,  by  clearing  of  fractions,  transposition,  and 
uniting,  as  the  particular  example  may  require,  be  united  into  one,  and  the 
results  arranged  in  the  order  given.  If,  then,  the  first  term,  i.  e.  the  one  con- 
taining the  square  of  the  unknown  quantity,  has  a  coefficient  other  than  unity, 
or  is  negative,  its  coefficient  can  be  rendered  unity  or  positive  without  destroy- 
ing the  equation  by  dividing  both  the  members  by  whatever  coefficient  this  term 
may  chance  to  have  after  the  first  reductions.     The  equation  will  then  take  the 


AF^cfflS   QUADRATICS.  131 

form  x^  ±  ax  =  ±  b.     Now  adding  ( -^  )  to  the  first  member  makes  it  a  perfect 

square  (the  square  ot  x  ±  ~j  ,  since  a  trinomial  is  a  perfect  square  when  one  of 

its  terms  (the  middle  one,  ax,  in  this  case)  is  ±  twice  the  product  of  the  square 
roots  of  the  other  two,  these  two  being  both  positive  {110,  Part  I,).  But,  if  we 
add  the  square  of  half  the  coefficient  of  the  second  term  to  the  first  member  iaZZZ 
make  it  a  complete  square,  we  must  add  it  to  the  second  member  to  preserve  the 
equality  of  the  members.  Having  extracted  the  square  root  of  each  member, 
these  roots  are  equal,  since  like  roots  of  equals  are  equal.     Now,  since  the  first 

term  of  the  trinomial  square  is  x^,  and  the  last   f  —  |  does  not  contain  x,  its 

square  root  is  a  binomial  consisting  ot  x  ±  the  square  root  of  its  third  term,  or 
half  the  coefficient  of  the  middle  term,  and  hence  a  known  quantity.  The 
square  root  of  the  second  member  can  be  taken  exactly,  approximately,  or  indi- 
cated, as  the  case  may  be.  Finally,  as  the  first  term  of  this  resulting  equation 
is  simply  the  unknown  quantity,  its  value  is  found  by  transposing  the  second 
tenn. 

Sen.  1. — This  process  of  adding  the  square  of  half  the  coefficient  of  tlie 
first  power  of  the  unknown  quantity  to  the  first  member,  in  order  to  make 
it  a  perfect  square,  is  called  Completing  the  Square.  There  arc  a  variety 
of  other  ways  of  completing  the  square  of  an  affected  quadratic,  some  of 
which  will  be  given  as  we  proceed ;  but  tliis  is  the  most  important.  This 
method  will  solve  all  cases:  others  arc  mere  matters  of  convenience,  in 
special  cases. 

Ill,  Cor.  1. — An  affected  quadratic  equation  has  two  roots. 
These  roots  may  both  be  positive,  both  be  negative,  or  one  positive  and 
the  other  negatiiie.      They  are  both  real^  or  both  imaginary, 

Dem. — Let  x"^  +  j).r  =  q  be  any  affected  quadratic  equation  reduced  to  the  form 
for  completing  the  square.     In  this  form  p  and  q  may  be  either  positive  or 

V) 

negative,   integral   or   fractional.      Solving   this   equation   we   have   a?  =  —  ^ 

±  \  -r-  ■\-  q.  We  will  now  observe  what  different  forms  this  expression  can 
take,  depending  upon  the  signs  and  relative  values  of  'p  and  q. 

Ist.   Wke)i\}  and  q  are  both  positiDC.     The  */////«  will  then  stand  as  given  ;  i.f., 

9^—  —~  ±  i/^  +  q.     Now,  it  is  evident  that    4/  ^'-  +  «/  >  V'    ^^^    V        +  7 

IS   the   square   root   of    something  more    than  ^-.      Hence,  as    ^  <  4/  -r-  +  q, 

4  ,0        '     4 

V  /  v'^  "0  /  'ft' 

—  ^  +  y  -^  +  <?  is  positim  ;    but  —  ^  —  y  ^  +  q  '^^  negative,  for  both  i)arts 

are  negative.  Moreover  the  negative  root  is  numerically  greater  than  the 
positive,  since  the  former  is  the  numerical  sum  of  the  two  parts,  and  the  latter 


132  ELEMENTARY  ALGEBRA. 

the  numerical  difference.  .'.  When  jp  and  q  are  both  +  in  the  given  fonn,  one 
root  is  positive  and  the  other  negative,  and  the  negative  root  is  numerically 
greater  than  the  positive  one. 

2d.  Wien  p  is  negative  and  q  positive.  We  then  have  x  = ~  ±  y  — ^  -f  q 

=  ~  ±  i/  ~  +  q.    If  we  take  the  plus  sign  of  the  radical,  x  is  positive  ;  but  if 

/p*  p 

we  take  the  —  sign,  x  is  negative,  since  y   ^  +  ?  >  o*     Moreover,  the  positive 

root  is  numerically  the  greater.  .'.  When  p  is  negative  and  q  positive,  one  root 
is  positive  and  the  other  negative  ;  but  the  positive  root  is  numerically  greater 
than  the  negative. 

3d.  When  p  and  q  are  both  negative.  We  then  have  x— ^ ±  y  ^—?—  +  (— y) 

=z  ^±  y  — —  q.    In  this  \i  ^  >  q,     y   j^  —  q  '^s  real, and  as  it  is  less  than  — , 
3       '     4  4  4  2 

V'  /  P* 

both  values  are  positive.     It  ^  =  q,    y    t —  g  =  0  and  there  is  but  one  value 

of  X,  and  this  is  positive.  (It  is  customary  to  call  this  two  equal  positive  roots 
for  the  sake  of  analogy,  and  for  other  reasons  which  cannot  now  be  appreciated 

by  the  pupil.)  If  ^  <  g,  y  L-.  —  q  becomes  the  square  root  of  a  negative 
quantity  and  hence  imaginary. 

4th.   When  n  is  positive  and  q  negative.    We  then  have  x  =  —  ~  ±  y  K —  q. 

As  before,  this  gives  two  real  roots  when  g  <  4-     ^^'^^^  ^^^Js  is  the  case  both 

roots  arc  negative.     [Let  the  pupil  show  how  this  is  seen.]     When  q:=~,  the 

roots  are  equal  and  negative ;  i.  e.,  there  is  but  one.  When  ^  <q  both  roots 
arc  imaginary. 

112,  Cor.  2. — An  affected  quadratic  being  reduced  to  the  form 
x2+  px  =  q,  the  valne  of  x  can  always  be  tcritten  out  without  taking 
the  intermediate  stejys  of  adding  the  square  of  half  the  coefficient  of 
the  second  term^  extracting  the  root.,  and  transposing.  Hie  root  in 
such  a  case  is  half  the  coefficient  of  the  second  term  taken  with  the 
opposite  sign,  ±  the  square  root  of  the  sum  of  the  square  of  this 
half  coefficient,  and  the  know7i  term  of  the  equation.     This  is  observed 

directly  from  the  form  x=  —  2±i/^+q^  and  more  in  detail 
in  the  demonstration  of  the  preceding  corollary. 


AFFECTED   QUADRATICS.  133 

H3»  Cor.  3. —  Upon  tJie  principle  that  the  middle  term  of  a  tri- 
nomial square  is  twice  the  product  of  the  square  roots  of  the  other 
two,  we  can  often  complete  the  square  more  advantageously  than  by 
the  regular  rule. 

Thus  having  4j;-  —  Vlx  =  10.  Since  4^^  is  a  perfect  square,  and  12iC  is  diria- 
ible  by  twice  the  square  root  of  4tx^ ,  i.  e.  by  4r,  we  see  that  the  wanting  third 
term  is  3',  or  9.     Adding  this  to  both  members,  we  have  4a;*—  VZx  +  0  =  25, 

Again,  if  the  coetficient  of  X'  is  not  a  perfect  square,  it  can  be  rendered  such 
by  multiplying  by  itself  (or  often  by  some  other  factor).  If  then  the  second 
term  (the  term  in  .r)  is  not  divisible  by  twice  the  square  root  of  this  first  term,  we 
may  multiply  both  members  of  the  equation  by  4,  and  the  first  term  will  still 
be  a  perfect  square,  and  the  second  tenn  divisible  by  twice  its  square  root. 

114*  ScH.  2. — The  method  of  Art.  110  is  perfectly  general,  and  will 
solve  all  cases  ;  but  some  may  prefer  the  more  elegant  methods  indicated  in 
{1 13),  in  special  cases.  Same  illustrations  of  these  methods  are  given  in 
the  examples  following. 


EXA3IPLE8. 

1.  x^-Qx=U.  2.  3.r2=24.'c-36.  3.  x^-^ax-la^. 

4.  x^-lx  +  2  =  10.     5.  3.^3+  135=12.7:.  6.  x^  +  {a-i)x^a. 
X        3     x—l            4x        x—1  a^x^     2ax    b^ 

=  -:-| Zi — •       t).  ; — -  — — r  —  4.  y.     ■  ,-- \-—r^=\J, 


.-c  -  1     2        2  x-\-  1     2x  +  'd  b^         a       c 

10.  Solve  9:c2+12a;=32,  7x^-Ux=  -of,  and  3x^-V3x=-.lO, 
by  Art.  li:i. 

Bug's. — Dividing  12.t'  by  2y^Qx',  or  Gx,  we  have  2  as  the  square  root  of  the 
third  term.  Hence  9.c'  +  12.r  +  4  =  86,  is  the  equation  with  the  square  com- 
pleted. 

7x*—  14j;  =  —  ;■>?,  becomes,  by  multiplying  by  7,  40.c'^—  98.r  =  —  40.  Hence, 
completing  the  square  as  in  the  last,  49x-—  9Sx  +  49  =  9. 

dx^  -  13.C  =  10,  multiplied  by  3  and  by  4  becomes  mx^  -  156a;  =  120.  Hence, 
completing  the  square  as  b(!fore,  36^''—  150.1'  +  (13)*=  289. 

[Note. — Solve  the  following  by  any  of  the  preceding  methods,  according  to 
taste  or  expediency.] 

11.  (2.r  +  3)^x(3a;4-7)*=12.  12.  3:^:2+2^^=85. 
13.  a^l+b^x^)  =  b(2a^x^{-b).                  14.  5x^ -~9x-^2i=0. 


15.  3Vll2-8.2;  =  19  +  \/3:c  +  7.  16.  7a;2-lU  =  6. 


134  ELEMENTARY   ALGEBRA. 

17.  (x-c)Vab-(a-b)\/(^=0.  18.  3x^+x=\L 


19.  5(?£Z1) ^  l^=Wx,  20.  ^^V-'-l^^_^ 

l  +  sVa^      Va  x—Vx^—a'^     ^ 

22.  ^-^^^      '^ 


1  +  Vl+^    l-Vl-a;*  .r+V.'c+i     11 

/,,_j     ,\fa-U     ,\     ^2  90      90         27 


25. 


4/4  +  V2a-3+a:«=^.  26.  2V^+ -— ^5 


■}  5  x—'Za      x  +  a 


29. 


31. 


X  +  V  a:^  —  9 


2Vx  +  ^4a:+VV^T~2=l.  30.  ;;'^  ^  ■"  =  (:. -2)  »> 


x^         ,  4     ,4.  1 


-(a^-b^)x= 


a^  +  h^  («*«)'*  +  (a«6)"i 


SECTION  IIL 

EQUATIONS  OP  OTHER  DEGREES  WHICH  MAY  BE   SOLVED  AS 

QUADRATICS. 

115^  Vrop^  l*^Anii  Pare  Equation  (i.  e.,  one  containing  the 
finknoicn  quantiti/  affected  i/^ith  hxtt  o?ie  exponent)  can  be  solved  in 
a  manner  sindkir  to  a  Pure  Quadratic. 

Dem.— In  any  such  equation  we  can  find  the  vahie  of  the  unknown  quantity 
affected  by  its  exponent,  as  if  it  were  a  pimple  equation.  If  then  the  unknown 
quantity  is  affected  witli  a  positive  integral  exponent  it  ran  be  freed  of  it  by 
evolution  ;  if  its  exponent  be  a  positive  fraction  it  can  be  freed  of  it  by  extract- 
ing the  root  indicated  by  the  numerator  of  the  exponent,  and  involving  this  root 
to  the  power  indicated  by  the  denominator.  If  the  exponent  of  the  unknown 
quantity  is  negative  it  can  be  rendered  positive  by  multiplying  the  equation  by 
the  unknown  quantity  with  a  numerically  equal  positive  exponent.     Q.  K.  D. 


HIGH]- 11   EQUA'JIONS    SOLVED   AS    QUADRATICS.  135 

Hff,  I*rop,  2\ — -^^'y  equation  containing  one  imknoicn  quan- 
tity affected  with  only  tico  different  exponents,  one  of  rohich  is  twic^ 
the  other,  can  be  solved  as  a)i  Aff'ected  Quadratic. 

Dem. — Let  m  represent  any  number,  positive  or  negative,  integral  or  frac- 
tional ;  then  the  two  exponents  will  be  represented  by  m  and  2m ;  and  the 
equation  can  be  reduced  to  the  form  x^"'  +  px"^  =  q.    Now  let  y  =  x'",  and  y'^=x^", 

whatever  m  may  be.     Substituting  we  have  y^  +  py  —  q,  whence  y  =  —  ^ 

^_1 

ir4/—  +  q.    But  y  —  a;"*;  hence  ^  =  (—  f  ±  y  x  "^  *^)   '    Q-  ^-  ^ 

117.  Prop,  fV. — Equations  may  frequently  he  put  in  the  form 
of  a  quadratic  by  a  judicious  grouping  of  terms  containing  the 
unknown  quantity,  so  that  one  group  shall  be  the  square  root  of  the 
other. 

Dem. — This  proposition  will  be  established  by  a  few  examples,  as  it  is  not  a 
general  truth,  but  only  points  out  a  special  method. 

lis.  Cor. — The  form  of  the  compound  iiY.wsi  may  sonieUmes 
be  found  by  transpositig  all  the  terms  to  the  first  member,  arranging 
them  with  reference  to  the  unknown  quantity,  and  extracting  the 
square  root.  In  trying  this  expedient,  if  the  highest  exponent  is 
not  eve?i  it  must  be  made  so  by  midtiplying  the  equation  by  the 
unknown  quantity.  In  like  mariner  the  coefficient  of  this  term  is 
to  he  made  a  perfect  square.  When  the  process  of  extracting  the 
root  terminates,  if  the  root  found  can  be  detected  as  a  part,  or  factor, 
or  factor  of  a  part  of  the  remainder,  the  root  may  he  the  polynomial 
term. 


110,  Prop,   4, —  Wheji  an  equation  is  reduced  to  the  form 

x"  +  Ax"~*  +  Bx"~'  +  Cx"~' f-  L  =  0,  the  roots  xcith  their  signs 

changed  are  factors  of  the  absolute  {knoion)  term  L. 

Dem. — 1st.  The  equation  being  in  this  form,  if  rt  is  a  root,  the  equation  is 
divisible  by  x  —  a.  For,  suppose  upon  trial  x  —  <i  goes  into  the  polynomial 
«"  +  A.c""''-^,  etc.,  Q  times  mth  a  remainder  R.  (Q  represents  any  series  of 
terms  which  may  arise  from  such  a  division,  and  R,  any  remainder.)  Now,  since 
the  quotient  multiplied  by  the  divisor,  +  the  remainder,  equals  the  dividend,  we 
have  (^  —  rt)  Q  +  R  =  a;"  +  A.r«-'  +  Bij"-'  +  C^;"-"  -  -  -  +  L.  But  this  polyno- 
mial =  0.  Hence  (.c  —  rt)Q  +  R  =:  0.  Now,  by  hypothesis  a  is  a  root,  and  conse- 
quently X  —  a  =  0.    Whence  R  —  0,  or  there  is  no  remainder, 

2d.  If  now  X  —  a  exactly  divides  a;"  +  Ax"*  ~  *  +  B.?;"  ~ '  +  Ct;"  —  ^  -  -  -  +  L,  a 
must  exactly  divide   L,  as   readily  appears   from  considering  the  process  of 


136  ELEMENTARY  ALGEBRA. 

division.      Hence  —  a  is  a   factor    of    L,   a  being  a   root  of    the    equation. 
Q.  E.  D. 

±20*  Many  equations  of  other  degrees  than  the  second,  and 
which  do  not  fall  under  the  preceding  cases,  may  still  be  solved  as 
quadratics  by  means  of  Special  Artifices.  For  these  artifices  the 
student  must  depend  upon  his  own  ingenuity,  after  having  studied 
some  examples  as  specimens.  These  methods  are  so  restricted 
and  special  that  it  is  not  expedient  to  classify  them ;  in  fact, 
every  expert  algebraist  is  constantly  developing  new  ones.  See 
Ex's.  47-57.  The  following  principle  is  often  of  service  in  such 
solutions : 

121,  Prop,  5, —  When  an  eqtiation  can  he  put  in  such  a  form 
that  the  product  of  any  number  of  factors  equals  0,  the  equation  is 
satisfied  by  putting  any  07ie  of  these  factors  equal  to  0. 

Dem. — This  scarcely  needs  demonstration,  but  will  appear  evident  if  we 
consider  such  an  expression  as  {x^  +  1)  («*—«'  +  1)  («—!)  =  0.  Now,  on  the 
hypothesis  that  any  factor,  as  a;'  +  1,  is  0,  the  equation  is  satisfied  *  So  also,  if 
a;'  —  «'  +  1  =  0,  the  equation  is  satisfied,  etc. 

122,  ScH. — Ability  to  recognize  a  factor  in  a  polynomial  is  of  prime  im- 
portance in  the .  soltUion  of  such  equations.  It  is  the  grand  key  to  difficult 
solutions. 


Examples. 

1.  x^  =  81.                   2.  x^  =  32.  3.  x^  =  m» 

4.  y^  =  243.                 5.  z^  ^  1331.  6.  y^  =  4. 

7.  ic"  =  b.            8.  x^+V^=  -    ^        .  9.  x^  -f-  ix^  =  12. 

x^-V-'i 

10.  x''^  -hx"'  -p,         11.  x^  -  x^-  56.  12.  ax^  +  hx^  =  c. 

13.  a^»  -  2ax^=  b.        14.  x^  +  x^  =  756.  15.  x^  4-  6x^-  22  =  0, 

16.  ax^  -hx^  -^c^O.  17.  .T*  +  -^  =  3f 


2x^ 


*  In  etrioti.css  wc  ehoald  add  "  »ince  this  hypothesis  cannot  render  any  other  factor  «. 


HIGHER  EQUATlONe   SOLVED  AS   QUADRATICS.  137 


18.  Sx*  -v/^  +  T-=i  =  16.       19.   x^  -  2x  +  Wx^  -2x  +  b  =  11.* 


20.  X  -\-W-  7Vx  +  IG  =  10  -  Wx  +  16. 


21.  a;2  -  a;  +  5  V2a;2  _  5^  +  6  =  ^(Sa;  +  33). 


22.  Vx+l'Z  +  ^/x  +  12  =  6.  23.  V  ^ -\-yi---x, 

X  '^ 

24.   1  +i/7^-  =  l/l+-.       25.  x^  +-,   +  2/'a:  +  -) 
X  a  x^  \        xj 


x 

142 
^9  * 


26.  2a:2  _  o^.  4.  2V2a;2  -  Ix  +  6  =  5a;  -  6. 


27.  7(1  +  :i:)2 -  a/(i  -  ^)'  =  yi  -  ^'.t 

28.  :c*  -  8a:3  4.  29a;2  -  52a:  +  36  =  126. 

Solution. —  See  {118).  Transposing  126,  and  extracting  the  square 
root ;  when  we  have  the  two  terms  'x^  —Ax  of  the  root,  we  have  a 
remainder  13.C-  —  52a;  —  90.  We  now  notice  that,  if  we  call  4  the  next  term 
of  the  root,  the  next  remainder  will  be  5^*  —  20a;  —106,  which  we  may 
write  5(a;*  —  4t'  +  4)  —  126.  Hence  our  equation  may  be  put  in  the  form 
(a;2  _  4p  +  4)2  4.  5(a;«  _  4^.  _,_  4)  ^  126. 

29.  a;*-6x3+5a;2+12a;=60.  30.  .T3-6a;8 +lla;=6. 
31.  4.t*  +  |=4.t3+33.  32.  a;^  4-5a;2 +3a:-9=:0.j: 
33.  .^3-63:2 +  13.1-10  =  0.  34.  a;3-13a:2 4-49.^-45  =  0. 
35.  a;3+8a;2  4-17a;  +  10=0.  36.  a;3  -  29.1-2 -f  198a: -360  =  0. 
37.  a:3-15.T2  +  74a:-120  =  0.  38.  a:*  +  2a;3-3a:2 -4.r+4=0. 


*a;«-2a;+  5  +  6  Va;2-2a;  +  5  =  16.  Pnttinga;*- 2a;  +  5  =  y^  y«+6y  =  16. 
Such  pnb^tituHon  is  not  absolutely  necessary,  as  we  may  treat  a;*  —  2a;  +  5  as  the  unknown 
quantity  without  substituting.    Solve  the  followinj?  in  like  manner. 

t  Dividing  by    1^\  _  x'i    "c  have    i /I  +  ^  _  a/^  —^  _  \     Then,  multiplyinir  by 

y   \—x       rl-i-a; 

%  By  (i/9)  we  arc  led  to  try  +  1  or  —  1,  or  +  or  —  3,  as  roots.  The  equation  is  divisible 
by  a:  —  1,  and  a;  +  3. 


138  ELEMENTARY  ALGEBRA. 

39.  x*'-10x^  +  35x^-50x  +  2i:=0.  40.  x*-4:X^ +Sx^-8x=21. 
41.  a;*-2a:3-25a;2+26a:=-120.  42.  3x* +  13a;3-117a;=243. 
,^  X       30   VZ  +  ix       1      ... 

U.x=^^  +  ^f.    PntVx=i,. 

X  —  D 


Special  Expedients. 

45.  To  find  the  roots  of  a:«  =  =t  1,  x^  =  ±il,  x*  =  =fc  1,  x^  =  ±l, 
x^  =  ±:lf  and  a;®  =  ±  1. 

Sug's.    a;'  —  1  =  0.    Factoring  {x  —  1)  (a;*  +  a; '  +  jc*  +  .r  +  1)  =  0.    .*.  x  =  l, 
and  also  a;*  +  a;'  +x*  +  x+  1  =  0.    Dividing  by  a;*,  a;'+ a;  +  1 +-  + -^  =0,or 


+  2  +  ^  +  *+i  =  i--("  +  D'^("  +  5)  =  ^- 


1  -f  x^ 
46.  To  find  the  roots  of  7^— — rr  =  «. 
(1  +  xy 


Sug's.    1  +  a;*  =  a(l  +  a;)*  =  a(l  +  4r  +  6a;'  4-  4r=»  4-  x*).    Whence,  dividing 

.       ,        .              .                       ,1         4a  /        1\        6rt 
by  a;*    and  arranging  terms,    x*  -\ — ,  — [x  +  -  I  =     _    . 


47.  To  solve  -=;  —  a  +  J. 

1  —  .r  +  V 1  -j-  x^ 

Sug's. — This  can  be  cleared  of  fractions,  and  tlien  of  radicals,  in  the  ordinary 

way.     But   the  following  expedient  will  be  found  elegant  in  this  case,  and 

convenient  in  many.     Dividing  by  2a,  treating  the   resulting  equation  as  a 

1—  x         a  —  b        1+a;*—  2a; 

proportion,  and  taking  it  by  division,  we  have      . = .'. ; — - 

^        ^  Vl+x'      a  +  b  1+x^ 

=  f ^^  ' ,  or  -?^  =  1  -  C^^) '  =     ^  ,.    Taking  this  again  by  com- 
Va  +  6/   '      1  +  a:^  \a  +  b/        (a  +  b)^  ^  ^ 

^   ,.   .  .  ,     .    (\+xY      (a  +  b)'+^ab      (a-b)^ +Sab 

position   and  division,  we  obtain  ^^— ^  =  -__^.^^_^  =.  _______  or 

l+a;       v/ (a  —  by  +  Sab  ..  -u^  . 

^— -;-  =r  — ^- .  Again,  by  division  and  composition,  we  obtain 

Via^^^Y  +Sab-(a-b) 


V{a  -by  +  Sab  +  {a-b) 


HIGHER  EQUATIONS  SOLVED  AS  QUADRATICS.  l3& 

48.  To  solve  (I  +  x  -{-x^)^  =  ^^-  (1  -{-  x^  -^  x^). 

SuG's.— Dividing  by  1  +«  +  a;^  1  +  a;  +  a;^  =  --±i  (1  -  a;  +  x\  or  l±^±f! 

a  —  1  X—x-^T? 

= .    :.  =  a. 

a  —  \  X 

49.  To  solve  a  =  a;^  +  (1  —  «)*. 

SuG's.— Since  (1— a;)*  =  (a;— !)*,■  we  may  write  a  =  («  —  i  +  i)*  +  (a;  —  i  —  ^)*. 
Now  put  X  —  ^  =  1/,  substitute  and  expand. 


50.  To  solve  \/x--  -  \/l--  =  ^— i 


XXX 


j's.— Dividing  by  i/ 1  _  1,  a/.^  +  1  - 1  =  ^'^.-     •     Squaring,  etc.,  2 y^a;  + 1 
f  X  y  X 

=  1  H h  ar.     Squaring,  etc.,  again,  (x j    —  2(x J=:  —  1. 


51.  Solve  x^  -x-h  SVTx^  -  3a;  +  2  =  -  +  7. 

9 

52.  Solve T  —  o  —  x  —  x^, 

1  -^  X  -\-  x^ 

53.  Solve  — ; — -  =  —  . 

a^  —  ax  +  x^       x^ 

54.  Solve  —  -_     —  =  Vx^  —  a*  (Va;*  +  Oic  —  V^^  —  ar). 

|/a;  +  Va;2-«2 


1  +  rr3         13 


55.  Soif6  2iV  1  -  a;^  =  a(l  +  icM.     Also   ,,         ,     _       . 

56.  Solve  6x^  —  5x^  -\- x  =  0.     Also  .^3  +  x^  —  4x  —  4  =  0. 

57.  Solve  8^:3  +  16a;  =  9.     Also  3x^  +  8.^*  —  8a;2  =  3. 

SuG. — The  solutions  of  the  last  four  depend  upon  the  recognition  of  a  com- 
mon factor. 


140  ELEMENTARY   ALGEBRA. 


SECTION  IV. 

SIMULTANEOUS  EQUATIONS  OF  THE  SECOND  DEGREE  BETWEEN 
TWO  UNKNOWN  QUANTITIES. 

123.  Prop,  1, —  Two  equations  between  two  imknoion  quanti- 
ties^ otie  of  the  second  degree  and  the  other  of  the  Jirst,  may  always 
he  solved  as  a  quadratic. 

Dem.— The  general  form  of  a  Quadratic  Equation  between  two  unknown 
quantities  is 

ax^  +  hxy  +  cy'^  -\-  dx  +  ey  ^f  =0, 

since  in  every  such  equation  all  the  terms  in  a;*  can  be  collected  into  one,  and  its 
coefficient  represented  by  a ;  all  those  in  xy  can  also  be  collected  into  one,  and 
its  coefficient  represented  by  6,  etc.,  etc. 

The  general  form  of  an  equation  of  the  First  Degree  between  two  variables  is 
a'x  +  h'y  +  c'  =  0. 

Now,  from  the  latter  x  = —, .  which  substituted  in  the  former  gives 

a 

no  term  containing  a  higher  power  of  y  than  the  second,  and  hence  the  resulting 

equation  is  a  quadratic.     Q.  E.  D. 


124:,  JProj),  2, — In  general^  the  solution  of  two  quadratics 
between  two  xinknown  quantities^  requires  the  solution  of  a  biqua- 
dratic ;  that  is,  an  equation  of  the  fourth  degree. 

Dem. — Two  General  Equations  between  two  unknown  quantities  have  the 
forms 

(1)  oar*  +  &j^  +  cy»  +  da;  +  «y  +/=  0,  and 

(2)  a'x"  +  h'xy  +  c'y^  +  d'x  +  e'y  +  /'  =  0. 
From  (1),  ^  =  -  ^  ±  /^^  -^yl±^y±f. 

Now,  to  substitute  this  value  of  x  in  equation  (2),  it  must  be  squared,  and 
also,  in  another  term,  multiplied  by  y,  either  of  which  operations  produces 
rational  terms  containing  y'^,  and  a  radical  of  the  second  degree.  Then,  to  free 
the  resulting  equation  of  radicals  will  require  the  squaring  of  terms  containing 
y',  which  will  give  terms  in  y*,  as  well  as  other  terms.     Q.  E.  D. 


12S,  Def. — A  Homogeneous  Equation  is  one  in  which 
each  term  contains  the  same  number  of  factors  of  the  unknown 


SIMULTANEOUS   QUADRATIC   EQUATIONS.  141 

quantities.     2x"  —  oxy  —  ?/2  =  16  is  homogeneous,     ^x^  —  ^^  +  ^^ 
=  10  is  not  homogeneous. 

120,  Prop,  3. — Two  Homogeneous  Quadratic  Equations  he- 
tween  two  unkno^cn  quantities  can  always  be  solved  by  the  method 
of  quadratics,  by  substituting  for  one  of  the  unknown  qua?itities  the 
product  of  a  new  unknown  quantity  into  the  other. 

Dem. — The  truth  of  this  proposition  will  be  more  readily  apprehended  by 

means  of  a  particular   example.       Take  the   two  homogeneous   equations  x'^ 

—  xy  +  y*  =  21,  and  y^  —  2xy  +  15  =  0.      Let  x  =  vy,  v  being  a  new  unknown 

quantity,  called  an  auxiliary,  whose  value  is  to  be  determined.     Substituting  in 

the  given  equations,  we  have  v^y^  —  vy^  +  y'^  =  31,  and  y'^  —  2vy^  =  —  15.    From 

21                               15 
these  we  find  y'^  =  —. 7 ,  and  y^  =  r .     Equating  these  values  of  y^, 

21  15 

— = ;  whence  42«  —  21  =  15^^  —  15i^  +  15.    This  latter  equation 

t)'  —  «  +  12«  —  1 

is  an  affected  quadratic,  which  solved  for  v,  gives  v  =  'S,  and  ^.    Knowing  the 

15 
values  of  v  we  readily  determine  those  of  y  from  y^  = ,   and  find  y 

=  ±  \/3  when  «  =  3,  and  y  =  ±5  when  v  =  ^.    Finally  as  x  =  vy,  its  values 
are  jc  =  ±  ^\^'S,  and  ±  4. 

By  observing  the  substitution  of  vy  for  x  in  this  solution,  it  is  seen  that  it 
brings  the  square  of  y  in  every  term  containing  the  unknown  quantities,  in  each 
equation,  and  hence  enables  us  to  find  two  values  of  y-  in  terms  of  v.  It  is  easy 
to  see  that  this  will  be  the  case  in  any  homogeneous  quadratic  with  two 
unknown  quantities,  for  we  have  in  fact,  in  the  first  of  the  given  equations,  all 
the  variety  of  terms  which  such  an  equation  can  contain.  Again,  that  the  equa- 
tion in  V  will  not  be  higher  than  the  second  degree  is  evident,  since  the  values  of 
y'^  consist  of  known  quantities  for  numerators,  and  can  have  denominators  of 
only  the  second,  or  second  and  first  degrees  with  reference  to  v.  Whence  v  can 
always  be  determined  by  the  method  of  quadratics  ;  and  being  determined,  the 

value  of  y  is  obtained  from  a  pure  quadratic  (y^  = — ,  in  this  case),  and  that 

2«  —  1 
of  X  from  a  simple  equation  {x=  vy  in  this  case). 


127,  JPvop,  4, —  When  the  unknoicn  quantities  are  similarly 
involved  in  two  quadratic,  or  even  higher  equations,  the  solution  can 
often  be  effected  as  a  quadratic,  by  substituting  for  one  of  the  un- 
known quantities  the  sum  of  two  others,  and  for  the  other  unknovm 
quantity  the  difference  of  these  new  quantities. 

As  this  is  only  a  special  expedient,  and  not  a  general  principle,  its  truth  will 
be  rendered  suflBciently  evident  by  the  solution  of  a  few  examples.  See  Ex's.  13, 
14,15. 


142  ELEMENTAKY  ALGEBRA. 

Examples. 

*    (5a;  +  2y=7.  *    t    x-{-y=2, 

(^+^=^'  .^(x^+y^=65, 

''      1  +  1=1.  'nxy=2S. 

^x    y 

^x^+xy=16,  (x*+xy-\-^*=e, 

j    rr2+  xy  +  2y^  =  7^,  ^     (X*+xy=12, 

i2x*+2xy-\-  y*=7'd.  '    \xy+y^=2. 

4?/2  =  9,  ^^     ix^+y^+l=Zxy, 


11. 


13. 


15. 


\xy  +  2y^=^.  '    \2{xy-^^)=^y*. 

(a:2+a:?/  +  ?/2  =  52,  j    a;«-2a:y-y8=31, 

\xy-x^  =  %.  '    {ix^+2xy-y^=101, 

U{x-\-y)=3xy,  ^^     cx^+y*=ixy, 

\x-\-y+x*-\-y^=2Q,  *    I    aj—^^  =ixy. 

(xy{x-\-y)=SO, 
(a:3 +2/8=35. 


SxjG.— The  last  three  are  readily  solved  by  (127)-  Thus,  in  the  15th,  putting 
jc  r=  3  +  T,  and  y  =  z  —  ti,  the  equations  become  2z^—  2v'z  =  30,  and  2z^  +  6«*2 
=  35. 

il«+/=20.  *-^+y^+-^=133.    ^'i.-,*  =  14560. 


Special   SoLUTioisfs. 

19.     yi  -  ^xy  +  20a;«  +  dy  -  264a;=0,         5y^  -  38a;^  +  x^  -  12^^ 
+  1056a;  =  0. 
Sua. — Add  4  times  the  first  to  the  second. 

*  Two  homogeneous  quadratics  can  always  be  solved  by  {126),  but  special  expedients  are  often 
more  ele;;ant.  In  this  case  by  adding  twice  the  second  to  the  first,  and  extracting  the  square 
root,  we  have  x  +  y-  ill.  Subtracting  twice  the  second  from  the  first,  and  extracting  the 
iquarc  root,  we  have   x—  y  z=.  ±3. 


SIMULTAKEOUS   QUADJIATIG   EQUATIONS.  143 

20.  X  +  y  =  x^,         '6y  —  x  —  y^. 
SuG. — Subtract  the  first  from  the  second. 

21.  r-^=^'         22.    ]^+^=''  23.1^+^=*' 

Sug's.— To  solve  the  23d,  square  the  first,  writing  the  result  ai^  +  2/*  =  16 
—  2a?y,  and  square  again.     Then  for  x^  +  y*  substitute  82. 

24.  To  solve  x  —  y  =  ^,  and  x^  —  y^  =z  3093. 

Sug's, — Divide  the  second  by  the  first,  and  proceed  in  a  manner  similar  to 
that  given  for  the  last. 

25.  To  solve  x^  —  xy  +  y^  =  7,  and  x*  +  x^y^  +  2/*  =  133. 
SuG. — Divide  the  second  by  the  first. 


26.  To  solve  (s  -  -^IL\  \  (s  +  -^l-) '  =  82,  and  xy  =  2. 

\        X  -\-  y/         \        X  —  y)  -^ 

SuG's.-Write  the  first  f?^zAn%    (^^±^\ ^  83.  and  put  ^^^^  =  ^. 
\  x^y  J  \  x-y  J  ^      x  +  y 

Whence  9^2  +  -^  =  82. 

27.  To  solve  x^  +  y  (xy  —  1)  =  0,  and  y^  —  x  (xy  +  1)  =  0. 

Sug's.— Write  x*  +  a;V  —  ^V  '—  0,  and  y*  —  a;V  —  xy=Q,  and  subtract  the 
second  from  the  first.     Whence  x*  —  y*  +  2x^y^  =  0,  or  x*  +2x^y^  +y*  =  2^*,  and 

a;*  +  y'  =  /y/2  y^,  or  -  z=  V\/2  —  1.    From  the  given  equations  we  get      ~^ 


xy 
,  .    Hence  .  ""  ^  =  3  -  2^/2.  or  xv  =  ^^72. 


^-    Hence^-^  =  3-2V2.or.Ty  =  iV2- 


28.  Given  xy  =  a{x  +  y),  xz  =  J(a;  4-  z)y  and  1/2;  =  c{y  +  z). 

SuG.— These  i^re  readily  put  into  the  forms  -  =  -  +  -,    -  =  -  +  -,  and  ^  =  - 

a     y     X     b      z     X    ^^    c      z 
1 
+  -. 

y 

29.  Given  x(x-{-y-\-z)  =  18,  y{x-hy+z)  =  12,  and  z{x-\-y+z)  =  6. 

30.  Given  xtjz  =  ^S,  ~  =  ^,  and  ^  =  t- 

^  yz       12  z        3 

31.  Given  x  +  y  +  z  =  6,  4:X  +  y  =  2z,  and  x^  +  ^8  +  ««  =  14. 

32.  Given  Wx^  -  y^  +  a^y  =  26,  and  7  -  |  =  ^0' 


144  ELEMENTARY  ALGEBRA. 

33.  Given  ^-^^  +  10^^^  =  1,  and  xy^  =  3. 

X- y  X  +  y 

34.  Given  y{x^  +  y^)  z^  4:{x  +  y)^,  and  xy  =  4(x  +  y), 

35.  Given  x  +  y  =  10,  and  a/-  +  i /^  =  -. 

36.  Given  Vi  —  v^  =  ^Vxy,  and  x  +  y  —  20. 


37.  Given  Vx^  +  y^  +  V^t^  -  y*  =  2y,  and  .t*  -  ^*  =  «*. 

38.  Given  A/-  +  i/ '-  =  -— :  +  1,  and  ^x^y  +  V^  =  78. 

Y    y        y    ^        yxy 

39.  Given  y a:  +  y  +  2Va:  —  V  =    \  ^  ,  and •—  =  —  . 

^/x-y  ^y  15 

'y2_64=8a:iy,        ,,      ( a:*  +  ?/^=3a:,         ._     (  8a:4-7/i=14, 


40.    -^^  :^'        41.    f \  '^,         '        42.    . 

(   y--4:=2y^x^,  (x^-{-y'^=x.  (    x^y^=2y*. 


Applications. 

1.  The  plate  of  a  looking-glass  is  18  inches  by  12,  and  it  is  to  be 
surrounded  by  a  plain  frame  of  uniform  width,  and  of  surface  equal 
to  that  of  the  glass.     Required  the  width  of  the  frame. 

2.  A  person  bought  some  fine  sheep  for  $360,  and  found  that  if  he 
had  bought  6  more  for  the  same  money,  he  would  have  paid  $5  less 
for  each.    How  many  did  he  buy,  and  what  was  the  price  of  each  ? 

3.  A  traveller  sets  out  for  a  certain  place,  and  travels  one  mile  the 
first  day,  two  the  second,  three  the  third,  and  so  on:  in  5  days  after- 
ward another  sets  out,  and  travels  12  miles  a  day.  How  long  and 
how  far  must  he  travel  before  they  will  come  together  ? 

4.  Divide  the  number  48  into  two  such  parts  that  their  product 
may  be  432. 

5.  Divide  the  number  24  into  two  such  parts  that  their  product 
may  be  equal  to  35  times  their  difference. 

6.  For  a  journey  of  108  miles,  6  hours  less  would  have  sufficed, 
had  the  traveller  gone  3  miles  an  hour  faster.  At  what  rate  did  he 
travel  ? 


APPLICATIONS   OF   QUADRATIC    EQUATIONS.  145 

T.  The  fore  wheel  of  a,  coach  makes  6  revolutions  more  than  the 
hind  wheel  in  going  120  yards;  but,  if  the  circumference  of  each 
wheel  be  increased  by  1  yard,  the  fore  wheel  will  make  only  4  revo- 
lutions more  than  the  hind  wheel  in  the  120  yards.  What  is  the  cir- 
cumference of  each  wheel  ? 

8.  The  product  of  two  numbers  isj!?;  and  the  difference  of  their 
cubes  is  equal  to  m  times  the  cube  of  their  difference.  Find  the 
numbers. 

9.  Find  two  numbers  whose  product  is  equal  to  the  difference  of 
their  squares,  and  the  sum  of  their  squares  equal  to  the  difference  of 
their  cubes. 

10.  There  are  4  numbers  in  arithmetical  progression.  The  sum  of 
the  extremes  is  8;  and  the  product  of  the  means  is  15.  What  are 
the  numbers  ? 

SuG. — In  solving  examples  involving  several  quantities  in  arithmetical  pro 
gression,  it  is  usually  expedient  to  represent  the  middle  one  of  the  series,  when 
the  number  of  terms  is  odd,  by  x,  and  let  y  be  the  common  difference.  If  the 
number  of  terms  is  even,  represent  the  two  middle  terms  by  x  —  y,  and  x  +  y, 
making  the  common  difference  2y. 

11.  Five  persons  undertake  to  reap  a  field  of  87  acres.  The  five 
terms  of  an  arithmetical  progression,  whose  sum  is  20,  will  express 
the  times  in  which  they  can  severally  reap  an  acre,  and  they  all 
together  can  finish  the  job  in  60  days.  In  how  many  days  can  each, 
separately,  reap  an  acre  ? 

12.  There  are  three  numbers  in  geometrical  progression,  the  sum 
of  the  first  and  second  of  which  is  9,  and  the  sum  of  the  first  and 
third  is  15.     Required  the  numbers.  ; 

Sug's.— In  solving  examples  involving  several  quantities  in  geometrical  pro- 
gression, it  is  sometimes  expedient  to  represent  the  tirst  by  x,  and  the  ratio  by  y, 
so  that  the  numbers  will  be  x,  xy,  xy^,  etc.  In  other  cases  it  is  expedient,  if  the 
number  of  numbers  sought  is  odd,  to  make  xy  the  middle  term  of  the  series  and 

V  x^  v^ 

'-  the  ratio.     Thus  5  terms  will  be  represented  -  ,  x^,  xy,  y-,  '—  .     When  the 

number  of  numbers  sought  is  even,  it  is  sometimes  expedient  to  represent  the 

two  means  by  x  and  v,  and  the  ratio  bv  -.     Thus  4  terms  become  —  ,  x,  y,  '—  . 
■'if'  '   X  y  X 

13.  There  are  three  numbers  in  geometrical  progression  wliose 
continued  product  is  64,  and  the  sum  of  their  cubes  is  584.  Required 
the  numbers.  :j.P 


146  ELEMENTARY  ALGEBRA. 

14.  The  sum  of  the  first  and  second  of  four  numbers  in  geometri- 
cal i)rogression  is  15,  and  the  sum  of  the  third  and  fourth  is  60. 
Required  the  numbers. 

15.  There  are  three  numbers  in  geometrical  progression,  whose 
product  is  64,  and  sum  14.    What  are  the  numbers  ? 

16.  It  is  required  to  find  four  numbers  in  arithmetical  progression, 
such  that  if  they  are  increased  by  3,  4,  8,  and  15  respectively,  the 
sums  shall  be  in  geometrical  progression. 

17.  It  is  required  to  find  four  numbers  in  geometrical  progression 
such,  that  their  sum  shall  be  15,  and  the  sum  of  their  squares  85. 

18.  The  sum  of  700  dollars  was  divided  among  four  persons.  A,  B, 
C,  and  D,  whose  shares  were  in  geometrical  progression ;  and  the 
difference  between  the  greatest  and  least,  was  to  the  difference  be- 
tween tlie  two  means,  as  37  to  12.     What  were  the  several  shares? 

19.  The  sum  of  three  numbers  in  harmonica!  proportion  is  191, 
and  the  product  of  the  first  and  third  is  4032 ;  required  the  numbers. 

20.  The  2d  and  6th  terms  of  a  geometrical  progression  are  respec- 
tively 21  and  1701.     What  is  the  first  term,  and  what  the  ratio  ? 

21.  A  and  B  travel  on  the  same  road,  at  tlie  same  rate,  and  in  the 
same  direction.  When  A  is  50  miles  from  the  town  D,  he  overtakes 
another  traveller  who  goes  at  the  rate  of  3  miles  in  2  hours ;  and 
two  hours  after,  he  meets  a  second  traveller  who  goes  at  the  rate  of 
9  miles  in  4  hours.  B  overtakes  the  first  traveller  45  miles  from  D, 
and  meets  the  second  40  minutes  before  he  (B)  reaches  the  31st  mile- 
stone from  D.     How  far  are  A  and  B  apart  ? 

22.  The  joint  stock  of  two  partners,  A  and  B,  was  $2080.  A's 
money  was  in  trade  9  months,  and  B's  6  months,  when  they  shared 
stock  and  gain,  A  receiving  $1140  and  B  $1260.  What  was  each 
man's  stock  ? 

23.  Tliere  is  a  number  consisting  of  three  digits,  the  first  of  which 
is  to  the  second  as  tlie  second  is  to  tlie  third;  the  number  itself  is 
to  the  sum  of  its  digits  as  124  to  7;  and  if  594  be  added  to  it  the 
digits  will  be  inverted.     What  is  the  number  ? 

24.  A  person  has  $1300,  which  he  divides  into  two  portions,  and 
loans  at  different  rates  of  interest,  so  that  the  two  portions  produce 


QUADRATIC    EQUATIONS.  147 

equal  returns.  If  the  first  portion  had  been  loaned  at  the  second 
rate  of  interest,  it  would  have  produced  136,  and  if  the  second  por- 
tion had  been  loaned  at  the  first  rate  of  interest,  it  would  have  pro- 
duced $49.     Required  the  rates  of  interest. 

25.  A  person  traveling  from  a  certain  place,  goes  1  mile  the  first  day, 
2  the  second,  3  the  third,  and  so  on ;  and  in  six  days  after,  another 
sets  out  from  the  same  place  to  overtake  him,  and  travels  uniformly 
15  miles  a  day.  How  many  days  must  elapse  after  the  second  starts 
before  they  come  together  ? 


148  ELEMEiiTAliY  ALGEBKA. 


CHAPTER  IV. 

INEQUALITIES. 

128,  An  Inequality  is  an  expression  in  mathematical  sym- 
bols, of  inequality  between  two  numbers  or  sets  of  numbers. 

Ill, — Thus  a  >  h  (read  "  a  greater  than  h ")  is  an  inequality  ;  also  a^x  —  3 
<  5  +  2  (read  "  a^x  -  3  less  than  5  +  2  ").     (See  Part  I.,  43,) 

129,  Fnndamental  Princijfle. — In  comparing  two  posi- 
tive numbers,  that  is  called  the  greater  which  is  numerically  so. 
Tlius  5  >  3.  But,  in  comparing  two  negative  numbers,  that  is 
called  the  greater  which  is  numerically  the  less.  Thus  —  5  <  —  3. 
Of  course  any  negative  number  is  less  than  any  positive  number.  In 
general,  we  call  a>h  when  a  —  h  \s  positive,  and  a  <h  when  a  —  h 
is  negative. 

130,  The  part  of  an  inequality  at  the  left  of  the  sign  >,  or  <, 
is  called  i\\^  first  memher^  and  the  part  at  the  right,  the  second  mem- 
ber of  the  inequality. 

131,  For  the  purposes  of  nuithematical  investigation,  inequali- 
ties are  subjected  to  the  same  transformations  as  equations,  but  with 
certain  characteristic  diiferences  in  the  results,  which  will  be  pointed 
out  in  the  following  propositions. 

132,  If,  in  transforming  an  inequality,  the  same  member  that 
was  the  greater  before  the  transformation  is  the  greater  after,  tlie 
inequality  is  said  to  continue  to  exist  in  the  same  sense ;  but,  if  tlie 
transformation  changes  the  general  relation  of  the  members,  so  that 
the  member  which  was  the  greater  before  the  transformation  is  the 
less  after,  tlie  inequality  is  said  to  exist  ill  an  opposite  sense  in  the 
two  inequalities. 

133,  JPvop, — The  sense  in  which  an  inequality  exists  is  not 
changed., 

1st.  By  adding  equals  to  both  member Sy  or  subtracting  equals  from 
both; 


INEQUALITIES.  149 

2d.  By  multiplying  or  dividing  the  members  by  equal  positive 
numbers  ; 

3d.  By  adding  or  multiplying  the  corresponding  members  of  two 
inequalities  which  exist  in  the  same  sense,  if  all  the  members  are 
essentially  positive  / 

4tli.  By  raising  both  tnembers  to  any  power  whose  index  is  an  odd 
number  ; 

5th.  £y  raising  both  members  to  any  power,  if  both  members  are 
essentially  ]?ositive  ; 

6  th.  By  extracting  the  same  root  of  both  members,  if  when  the  de- 
gree of  the  root  is  even,  only  the  positive  roots  be  compared. 

III.  and  Dem. — The  1st  is,  in  general,  an  axiom.  Thus  if  a  >  h,  it  is  evi- 
dent that  a  ±  c  >  h  ±  e.  When  c>  a,  <i  —  c'ls,  negative,  but  since  b  <  a,  b  —  c 
is  also  negative  and  numerically  greater  than  a  —  c.  Therefore,  in  this  case, 
a-ob  -c  {120). 

2d.  This  is  wholly  axiomatic.     If  a  >  &  it  is  evident  that  ma  >  mb,  and  that 

m      m'  . 

3d.  This,  too,  is  an  axiom.  \i  a  >b,  and  c>  d,  a,  b,  c,  and  d  being  each  + , 
it  is  evident  that  a  +  c  >  b  +  d;  and  that  ae  >  bd. 

4th.  This  becomes  evident  by  considering  that  if  a  >  b,  raising  both  members 
to  any  power  whose  degree  is  odd  will  leave  the  sigiis  of  the  members  as  at  the 
first,  and  also  the  sense  of  the  numerical  inequality  the  same. 

5th.  This  appears  from  the  fact  that  neither  the  signs  nor  the  sense  of  the 
numerical  inequality  of  the  members  is  changed  by  the  process. 

Cth.  This  is  evident  from  the  fact  that  the  greater  number  has  the  greater 
root,  if  only  positive  roots  are  considered. 


134,  Prop, —  The  8e?i8e  in  %ohich  an  inequality  exists  is  changed^ 

1st.  By  changing  the  signs  of  both  members ; 

2d.  By  mxdtiplying  or  dividing  both  members  by  the  same  negative 
quantity  ; 

3d.   By  raising  both  members  to  the  same  even  power,  if  the  members 
are  both  negative  in  the  first  instance  / 


150  ELEMENTARY  ALGEBRA. 

4th.  By  comparing  the  negative  even  roots  (the  members^  in  the  first 
instance,  being  both  essentially  positive), 

III,  and  Dem.— The  first  is  evident,  since  Si  a>h,    —aK  —  hhy  {129). 
That  is,  of  two  negative  quantities  the  numerically  greater  is  really  the  less. 

2d.  These  operations  do  not  change  the  numerical  relation  of  the  members, 
Ijut  do  change  the  signs  of  the  members  ;  hence  it  falls  under  the  preceding. 

3d.  and  4th.  Essentially  the  same  reasoning  as  in  the  last. 


Exercises. 

1.  When  a  and  b  are  unequal,  show  that  «*  +  b*>2ab. 

Solution. — Let  a>h;  whence  a—b>0,  or  a*—2ab  +  b^>0,oTa*  +  b*>2ab. 
Similarly  if  a  <b. 

2.  Prove  that  the  arithmetical  mean  between  two  quantities  is, 
in  general,  greater  than  the  geometrical.  Hoav  if  the  quantities  are 
equal  ? 

3.  If  «,  b,  c,  are  such  that  the  sum  of  any  two  is  greater  than  the 
third,  show  that  a^ -\- b^  +  c^  <2(ab  -\-  ac -h  be). 

4.  If  o*  +  J*  +  c^=  1,  and  m*  +  n^  +  r^=  1,  show  that  am  ^-bn 
+  cr  <  1.    How  \fa  =  b  =  c  =  m  =  n  =  r^ 

5.  Show  that,  in  general,  {a-\-b-c)?  +  {a-\-c—b)^-\-  (b  +  c— a)^ 
> ab •{■  be -\- ac.    How  if  a=b—c? 

G.  Which  is  greater,  2x^  or  .c  +  1  ? 

Solution.— 1st.  If  x>l,  a;*>  1(?),  2x^>2x{1);  but  2aj>«  +  l(?). 
,-.  3a;' >  X  +  1. 

If  X  <1,  a  similar  process  shows  2a;'*  <  a;  +  1. 

7.  From  5a;  —  6  <  3a:  +  8,  and  2a;  4-  1<  3a;  —  3,  show  that  x 
may  have  any  value  between  7  and  4 ;  i.  e.,  that  the  limiting  values 
are  7  and  4. 

8.  What  are  tlio  limiting  values  of  x  determined  from  the  con- 
ditions 3a;  —2  >  ^x  —  ^,    and    J  —  Ja;  <  8  —  2a;  ? 

9.  The  double  of  a  iiuml)er  diminished  by  o  is  greater  than  25, 
and  triple  of  the  number  diminished  by  7  is  less  than  tlie  double 
increased  by  13.     What  numbers  will  satisfy  the  conditions? 


PART    III. 


AN    ADVANCED    COURSE    IN 
ALGEBRA. 


CHAPTER  I. 

INFINITESIMAL  ANALYSIS. 


SECTION  I. 

DIFFERENTIATION. 

135,  In  certain  classes  of  problems  and  discnssions  the  quantities 
involved  are  distinguished  as  Constant  and  Variable. 

136,  A  Constant  quantity  is  oric  wliich  maintains  the  same 
value  throughout  the  same  discussion,  and  is  represented  in  the 
notation  by  one  of  the  leading  letters  of  the  alphabet. 

137,  Variable  quantities  are  such  as  may  assume  in  the  same 
discussion  any  value  within  certain  limits  determined  by  the  nature 
of  the  problem,  and  are  represented  by  the  final  letters  of  the 
alphabet. 

III.— If  X  is  the  radius  of  a  circle  and  j/  is  its  area,  p  =  7tx^,  as  we  learn  from 
Geometry,  7t  being  about  3.1410.  Now  if  a',  the  radius,  varies,  y,  the  area,  will 
vary  ;  but  it  remains  the  same  for  all  values  of  x  and  y.  In  this  case  x  and  y 
are  the  variables,  and  ;r  is  a  constant. 

Again,  if  y  is  the  distance  a  body  falls  in  time  x,  it  is  evident  that  the  greater 
X  is,  the  greater  is  y,  i.  e.,  that  as  x  varies  y  varies.  We  learn  from  Physics  that 
y  —  lQ^fX'^,  for  comparatively  small  distances  above  the  surface  of  the  earth. 
In  the  expression  y  =  16,^^3;'^,  x  and  y  are  the  variables,  and  16,\-  is  a  constant. 

Once  more,  suppose  we  have  y^  =  25x^  —  3.^;*  —  5,  as  an  expressed  relation 
between  x  and  y,  and  that  this  is  the  only  relation  which  is  required  to  exist 


152  ADVANCED  COURSE  IN  ALGEBRA. 

between  them ;  it  is  evident  that  we  may  give  values  to  x  at  pleasure,  and  thus 
obtain  corresponding  vahies  for  p.  Thus  if  x  =  l,  y  =  ±  V\l,  if  x  =  %,  y 
=  ±  Vi88,  etc.,  etc.  In  such  a  case  x  and  y  are  called  variables.  But  we  notice 
that  if  we  give  to  x  such  a  value  as  to  make  ^x^  +  5  >  25^'^  (as,  for  example,  ^, 
\,  etc.),  y  will  be  imaginary.  This  is  the  kind  of  limitation  referred  to  in  our 
definition  of  variables.* 

138,  ScH. — The  pupil  needs  to  guard  against  the  notion  that  the  terms 
constant  and  vnriaUe  arc  synonyms  for  known  and  unknown^  and  the  more  so 
as  the  notation  might  lead  him  into  this  error.  Tlie  quantities  he  has  been 
accustomed  to  consider  in  Arithmetic  and  Elementary  Algebra  have  all  been 
constant.  The  distinction  here  made  is  a  new  one  to  him,  and  pertains  to  a 
new  class  of  problems  and  discussions. 

139,  A  Function  is  a  quuntit}',  or  a  mathematical  expression, 
conceived  as  depending  for  its  valne  upon  some  other  quantity  or 
quantities. 

III. — A  man's  Avages/o?"  a  given  time  is  a  function  of  the  amount  received  per 
day,  or,  in  general,  his  wages  is  a  function  of  the  time  he  works  and  the  amount 
he  receives  per  day.  In  the  expression  y  =  16,V-c*  {Vi7),  second  illustration, 
y  is  a  function  of  x,  i.  e.,  the  space  fallen  through  is  a  function  of  the  time.  The 
expression  2«x*  —  3a;  -:-  5&,  or  any  expression  containing  x,  may  be  spoken  of  as 
a  function  of  x. 

140,  When  we  wisli  to  indicate  that  one  variable,  as  y,  is  a  func- 
tion of  another,  as  x,  and  do  not  care  to  be  more  specific,  we  write 
9/  =f(x),  and  read  "?/  equals  (or  is)  a  function  of  x"  This  means 
nothing  more  than  that  i/  is  equal  to  some  expression  containing  the 
variable  x,  and  which  may  contain  any  constants.  If  we  wish  to 
indicate  several  different  expressions  each  of  which  contains  a:,  we 
>vrite/(.T),  (p{x)y  orf'{x),  etc.,  and  read  "the /function  of  x,'*  "the 
(p  function  of  .r,"'  or  "  the/'  function  of  a*." 

III. — The  exprestion /(«)  may  stand  for  x^  —2x  +  5,  or  for  3(a'  —  x^),  or  for 
any  expression  containing  x  combined  in  any  way  with  itself  or  with  constants. 
But  in  the  same  discussion  f{x)  will  mean  the  same  thing  throughout.  So  again, 
if  in  a  particular  discussion  we  have  a  certain  expression  containing  x  (e.  </., 
3a!*  —  ax  +  2ab),  it  may  be  represented  by /(.t),  while  some  other  function  of  x 
(e.  g.,  n{a^  —  x"^)  +  2x'-)  might  be  represented  by /'(«),  or  q>{x). 

.,  141,  In  equations  expressing  the  relation  between  two  variables, 
as  in  ?/2  =:  3ax^  —  x^,  it  is  customnry  to  sjuak  of  one  of  the  variables, 
as  ?/,  as  a  function  of  the  other  .r.     Moreover,  it  i^  convenient  lo  think 


*  The  limits  of  this  vohime  do  not  permit  the  interpretation  of  imaginnrics  a«^  other  than  im- 
po?siblo  quantities,  i.  e.,  inconsistent  with  tiic  restricted  view  talicn  of  the  particular  problem 
which  may  be  under  conbidcration. 


DIFFERENTIATION.  153 

of  X  as  varying  and  thus  producing  change  in  y.  When  so  con- 
sidered, X  is  called  the  Independent  and  y  the  Dependent  variable. 
Or  we  may  speak  of  ^  as  a  function  of  the  variable  x, 

142,  An  Infinitesimal  is  a  quantity  conceived  under  such 
a  form,  or  law,  as  to  be  necessarily  less  than  any  assignable  quantity. 

Infinitesimals  are  the  increments  by.  which  continuous  number,  or 
quantity  (S),  may  be  conceived  to  change  value,  or  grow. 

III. — Time  affords  a  good  illustration  of  continuous  quantity,  or  number. 
Thus  a  period  of  time,  as  5  hours,  increases,  or  grows,  to  another  period,  as  7 
hours,  by  infinitesimal  increments,  i.  c,  not  by  liours,  minutes,  or  even  seconds, 
but  by  elements  which  are  less  than  any  assignable  quantity.  In  this  way  we 
may  conceive  any  continuous,  variable  quantity  to  change  value,  or  grow,  by 
infinitesimal  increments. 

14:3,  Consecutive  Values  of  a  function,  or  variable,  are 
values  which  differ  from  each  other  by  less  than  any  assignable 
quantity,  /.  e.,  by  an  infinitesimal  part  of  either. 

144=,  A  Differential  of  a  function,  or  variable,  is  the  difier- 
ence  between  two  consecutive  states  of  the  function,  or  variable.  It 
is  the  same  as  an  infinitesimal. 

III. — Resuming  the  illustration  y=r»6-,Vc'  (IST)*  let  x  be  thought  of  an 
some  particular  period  of  time  (as  5  seconds),  and  y  as  the  distance  through 
which  the  body  falls  in  that  time.  Also,  let  x'  represent  a  period  of  time  infini- 
tesimally  greater  than  x,  and  y'  the  distance  through  which  the  body  falls  in  time 
x'.  Then  x  and  x'  are  consecutive  values  of  .r,  and  y  and  y'  are  consecutive 
values  of  y.  Again,  the  difference  between  x  and  x',  as  x'  —  .?■,  i.,  a  differential 
of  the  variable  x,  and  y'—  y  is  n  differential  of  the  function  y. 

145,  W^otation,— A  differential  of  x  is  expressed  by  writing  the 
letter  d  before  x,  thus  dx.  Also,  dy  means,  and  is  read  ^*  differen- 
tial yr 

Caution. — Do  not  read  dx  by  naming  the  letters  as  you  do  ax ;  but  read  it 
•'  differential  x."  The  d  is  not  a  factor,  but  an  abbreviation  for  the  word  differ- 
ential. 

146,  To  lyifferentiate  a  function  is  to  find  an  expression 
for  the  increment  of  the  function  due  to  an  infinitesimal  increment 
of  the  variable;  or  it  is  the  process  of  finding  the  relation  between 
the  infinitesimal  increment  of  the  variable  and  the  corresponding 
increment  of  the  function. 


154  ADVANCED  COURSE  IN  ALGEBRA. 

Rules  for  Differentiating. 

147,  RULE  1. — To  differentiate  a  single  variable,  sim- 
ply  WRITE   the   letter   cl   BEFORE   IT. 

This  is  merely  doing  what  the  notation  requires.  Thus  if  x  and  x'  are  conse- 
cutive states  of  the  variable  x,  i.  e.,  if  x'  is  what  x  becomes  when  it  has  taken  an 
infinitesimal  increment, «'—  x  is  the  differential  of  x,  and  is  to  be  written  ci«.  In 
like  manner,  y'—  y  is  to  be  written  dy,  y'  and  y  being  consecutive  values. 


148.  RULE  2. — Constant  factors  or  divisors  appear  in 

THE    DIFFERENTIAL  THE  SAME  AS  IN  THE  FUNCTION. 

Dem, — Let  us  take  the  function  y  =  ax,  in  which  a  is  any  constant,  integral 
or  fractional.  Let  x  take  an  infinitesimal  increment  d.r,  becoming  x  -f  dx ;  and 
let  dy  be  the  corresponding  *  increment  of  y,  so  that  when  x  becomes  x  +  dx,  y 
becomes  y  +  dy.     We  then  have 

1st  state  of  the  function      -    -    -    - y  =  ax ; 

2d,  or  consecutive  state        y  +  dy  =  a(x  +  dx)  =  €ix  +  adx. 

Subtracting  the  1st  from  the  2d dy  =  adx, 

which  result  being  the  difference  between  two  consecutive  states  of  the  function, 
is  its  differential  {144:),     Now  a  appears  in  the  differential  just  as  it  was  in  the 

function.     This  would  evidently  be  the  same  if  a  were  a  fraction,  as  — ,     We 

should  then  have,  in  like  manner,  dy  —  —dx   as  the   differential  oi    y  =  —x. 
Q.  E.  D. 


149,  RULE  3. — Constant  terms  disappear  in  differen- 
tiating;  OR  THE   DIFFERENTIAL   OF   A   CONSTANT   IS   0. 

Dem. — Let  us  take  the  function  y  =z  ax  +  h,  in  which  a  and  h  arc  constant. 
Let  X  take  an  infinitesimal  increment  and  become  x  +  dx  ;  and  let  dy  be  the 
increment  which  y  takes  in  consequence  of  this  change  in  x,  to  that  when  x 
becomes  x  +  dx,  y  becomes  y  +  dy.     W^e  then  have 

1st  state  of  the  function y  =  flW!  +  &; 

2d,  or  consecutive  state       -    -    .    .     y  +  dy  =  a{x  +  dr)  +  &  =  ax  +  adx  +  h 

Subtracting  the  1st  from  the  2d       -     -    -    -  dy  z=  adx, 

which  being  the  difference  between  two  consecutive  states  of  the  function,  is  its 
differential  {144).     Now  from  this  differential  the  cons^tant  h  has  disappeared. 

We  may  also  say  that  as  a  constant  retains  the  same  value,  tlu  ro  is  no  differ- 


*  The  Word  "contemporaneous"'  is  often  ueed  iu  this  cunncction. 


DIFFERENTIATION.  155 

ence  between  its  consecutive  states  (properly  it  has  no  consecutive  states). 
Hence  the  differential  of  a  constant  may  be  spoken  of  (though  with  some  lati- 
tude) as  0.     q.  E.  D. 


150,  RULE  4. — To  differentiate  the  algebraic   sum    of 

SEVERAL  VARIABLES,  DIFFERENTIATE   EACH   TERM   SEPARATELY  AND 
CONNECT  THE  DIFFERENTIALS  WITH  THE   SAME  SIGNS  AS  THE  TERMS. 

Dem. — Let  u=ix-'fy  —  Zy  u  representing  the  algebraic  sum  of  the  variables 
X,  y,  and  —z.  Then  is  du  =  dx  ■\-  dy  —  dz.  For  let  dx,  dy,  and  dz  be  infinitesimal 
increments  of  x,  y,  and  z ;  and  let  du  be  the  increment  which  u  takes  in  conse- 
quence of  the  infinitesimal  changes  in  x,  y,  and  z.    We  then  have 

1st  state  of  the  function w  =  a;  +  y  —  2; 

2d,  or  consecutive  state u  ->t  du  —  x -v  dx  ^-  y  -v  dy  —  {z -\-  dz), 

Or u  +  du  —  x  +  dx+y+dy—z  —  dz. 

Subtracting  the  1st  state  from  the  2d      -    -    -   du  =  dx  +  dy  —  dz.     q.  e.  d. 


ISl,  RULE  5. — The  differential  of  the  product  of  two 

VARIABLES   IS  THE   DIFFERENTIAL  OF   THE   FIRST   INTO  THE   SECOND, 
PLUS   THE    DIFFERENTIAL   OF   THE    SECOND   INTO   THE    FIRST. 

Dem. — Let  u  =  xy  be  the  first  state  of  the  function.  The  consecutive  state  is 
u  +  du  =  {x  4-  dx){y  +  dy)  =  xy  +  ydx  +  xdy  +  dx  dy.  Subtracting  the  1st  state 
from  the  consecutive  state  we  have  the  differential,  i.  e.,  du  —  ydx  +  xdy  +  dx  dy. 
But,  &s  dxdy  is  the  product  of  two  infinitesimals,  it  is  infinitely  less  than  the 
other  terms  (ydx  and  xdy),  and  hence,  having  no  value  as  compared  with  them,  is 
to  be  dropped.*    Therefore  du  =  ydx  +  xdy.    q.  e.  d. 


1S2,  RULE  G. — The  differential  of  the  product  of  sev- 
eral VARIABLES  IS  THE  SUM  OF  THE  PRODUCTS  OF  THE  DIFFER- 
ENTIAL OF   EACH    INTO   THE   PRODUCT   OF   ALL  THE   OTHERS. 

Dem. — Let  u  =  xyz  ;  then  du  —  yzdx  +  xzdy  +  xydz.  For  the  1st  state  of  the 
function  is  w  —  xyz,  and  the  2d,  or  consecutive  state,  u  +  du  =  (x  +  dx)  {y  +  dy) 
(z  +  dz),    or  u  +  du  =  xyz  +  y^dx  +  xzdy  +  xydz  +  xdydz  +  ydxdz  +  zdxdy 


*  It  will  doubtle«K  appear  to  the  pnpil,  at  first,  as  if  this  gave  a  rcsuU.  only  approodinaleJy  rnr- 
rcct.  Such  is  not  tli?  fdct.  The  result  is  absoMdy  correct.  No  error  is  introduced  by  dropping 
dx  dy.  In  fact  this  term  mt'sf.  be  dropped  according  to  the  nature  of  infinitesimals.  Notice 
that  by  definition  a  quantity  which  is  iiidni^c^invil  with  respect  to  another  is  one  which  has  710 
assignable  magnihidc  with  reference  to  that  other.  Hence  wo  must  so  treat  it  in  our  re  isoiiing. 
Now  dxdy  ie  an  iufinitesimal  of  an  infinitcs'mnl  (t.  e..  two  infinitesimals  multipl'el  together), 
and  hence  is  infinitesimnl  with  reference  to  ijdx  and  x  "y.  and  mu^t  be  treated  as  having  no  as- 
signable v,ilnc  with  respect  to  them  ;  that  is.  it  must  bo  dropped. 


I5(j  ADVANCED  COURSE  IN  ALGERRA. 

■\-  (Udydz.     Subtracting,  and  (Iroi)ping  all  infinite»im-ila  of   infinitesimala  (see 
preceding  rule  and  f(x>t-note),  we  have  du  =  yzdx  +  xzdy  +  a-ydz. 

In  a  similar  manner  the  rule  can  be  demonstrated  for  any  number  of  varia- 
bles.    Q.  E.  D. 


153.  RULE  7.— The  differential  of  a  fraction   having 

A  VARIABLE  NUMERATOR  AND  DENOMINATOR  IS  THE  DIFFEREN- 
TIAL OF  THE  NUMERATOR  MULTIPLIED  BY  THE  DENOMINATOR, 
MINUS  THE  DIFFERENTIAL  OF  THE  DENOMINATOR  MULTIPLIED  BY 
THE   NUMERATOR,   DIVIDED   BY  THE   SQUARE   OF  THE   DENOMINATOR. 

X  vdx xdv 

Dem. — Let    u  =   -  ;    then   is    du  = :t — ~  .     For,  clearing  of  fractions. 

y  y*  ^ 

yu  =  X.  Differentiating  this  by  Rule  5th,  we  have  udy  +  ydu  =  d.r.  Substi- 
tuting for  u  its  value  '-,  this  becomes' — -  +  ydu  =  dx.     Finding  the  value  of 

-            ,           _         ydc  —  xdy 
du,  we  have   dn  = , -.      o.  e.  d. 

154,  Con. —  Tlie  differential  of  a  fraction  having  a  constant 
numerator  and  a  variable  denominator  is  the  product  of  ths  numera- 
tor with  its  sign  changed  into  the  differential  of  the  denominator,  di- 
vided by  the  square  of  the  denominator. 

Let  u  =  ■-.  Differentiating  this  by  the  rule  and  calling  the  differential  of 
the  constant  (a)  0,  wo  have  du  = -—  —  ^^—  —  .      o   v:  D 

liio.  Sen. — If  the  numerator  is  variable  and  tlie  denominator  constant, 
it  falls  under  Rule  2. 


150.  RULE  8.— The  differential  of  a  variable  affected 

WITH  AN  exponent  IS  THE  CONTINUED  PRODUCT  OF  THE  EXPO- 
NENT, THE  VARIABLE  WITH  ITS  EXPONENT  DIMINISHED  BY  1,  AND 
THE   DIFFERENTIAL   OF   THE   VARIABLE. 

Dem.— 1st.  When  the  exponent  is  a  positive  integer.  Let  y  =  of",  m  being  a 
positive  integer ;  then  dy  =  ntV'-^dx.  For  y  =  x"^  -  xxx-x-  to  m  factors.  Now, 
differentiating  this  by  Rule  G,  we  have  dy  =  (xxx  -  -  to  m  -  1  factors)  dx 
+  (xxx  .  .  to  m  —  1  factors)  dv  +  etc.,  to  m  terms ;  or  dy  =  .T^'-'dr  +  x'^-^dx 
+  x"'-^dx  +  etc.,  to  m  terms.     Therefore  dy  =  mx^-^dx. 

m 

2d.   When  the  cxjyoncnt  i^  a  positive  fraction.    Let  y  =  x'^,'^  being  a  positive 

7)1 

fraction  ;  then  dy  =  ^.i"  dx.  For  involving  both  members  to  tlie  nth  power  wo 
have  y"  =  x'".    Differentiating  this  as  just  shown,  we  Ijave  ny^-'^dy  ==  mx"i~'^dx. 


DIFFERENTIATION.  157 

^  win— m 

Now  from  y  =  ic»  we  have  y^-'  =  x~^ .     Substituting  this  in  the  last  it  be- 
comes  wic    "    dy  =  mx"'-Hx ;  vfhence  dy  =  '^x"  n    dx  =  ^x^     dx.    q.  e.  d. 

•    3d.   When  the   exponent    is    negative.      Let  y  =  .'c-'»,  n  being    integral    or 

fractional ;  then  dy  —  —  nx-'*-^dx.    For  y  =  a;-'*  =  — ,  which  differentiated  by 

X"  •' 

Rule  7,  Cor.,  gives  dy  = — —  =  —  nx-^'-^dx.    q.  e.  d. 


Examples. 

1.  Differentiate  //  =  'Sx-  —  2x  -{-  A. 

Solution.— The  result  is  dy  =  6vdx  -  2dx.  Which  is  thus  obtained  :  By 
Rule  1,  the  differential  of  y  is  dy.  To  differentiate  the  second  member  we  dif- 
ferentiate each  term  separately  according  to  Rule  4.  In  differentiating  3.i;^  we 
observe  that  the  factor  3  is  retained  in  the  differential,  Rule  2,  and  the  differen- 
tial of  .1*  is,  by  Rule  8,  2xd.r.  Hence,  the  differential  of  dx'^  is  6xdx.  The  differ- 
ential of  —  Zv  is  —  2dx.  By  Rule  3,  the  constant  4  disappears  from  the  differen- 
tial, or  its  differential  is  0, 

'Z.  Differentiate  y  =  2ax^  +  4:Clv^  —  x  +  m. 

Result,    dy  =  ^axdx  +  VZax^dx  —  dx> 

3.  Differentiate  y  =  Ur?  —  30.?:'  +  4.t. 

4.  Differentiate  y  =  Ax'  +  Bx'  +  Cx\ 

157.  'ScH. — It  is  desirable  that  the  pupil  not  only  become  expert  in  writ- 
ing out  the  differentials  of  such  expressions  as  the  above,  but  that  lie  know 
what  the  operation  signifies.  Thus,  suppose  we  have  the  equation  y  =  ^x. 
This  expresses  a  relation  bctAvcen  x  and  y.  Now,  if  x  changes  value,  y  must 
change  also  in  order  to  keep  the  equation  true.  In  this  simple  case  it  is  easy 
to  see  that  y  must  change  5  times  as  fast  as  x  in  order  to  keep  the  equation 
true.  This  is  what  differentiation  shows.  Thus,  differentiating,  we  have  dy 
=  bdx.  That  is,  if  x  takes  an  infinitesimal  increment,  y  takes  an  infinitesi- 
mal increment  equal  to  5  times  that  which  x  takes;  or,  in  other  words,  y 
increases  5  times  as  fast  as  x. 

Now  let  us  take  a  case  which  is  not  so  simple.  Let  y=Bx'^—2x+4:,  and  let 
it  be  required  to  find  the  relative  rate  of  change  of  x  and  y.  Differentiating, 
we  have  dy  =  6xdx  —  2dx  —  (6x  —  2)dx.  This  shows  that,  if  x  takes  an  infini- 
tesimal increment  represented  by  dx,  y  takes  one  (represented  by  dy)  which 
is  6a;  —  2  times  as  large ;  /.  e.,  that  y  increases  Qx  —  2  times  as  fast  as  x. 
Notice  that  in  this  case  the  relative  rate  of  increase  of  x  and  y  depends  on 
the  value  of  x.  Thus,  when  x=l,  y  is  increasing  4  times  as  fast  as  x  \  when 
x=2,  y\s  increasing  10  times  as  fast  as  a;;  when  a; =3,  y  is  increasing  Iff 
times  as  fast  as  x ;   etc. 


158  ADVANCED   COURSE  IN  ALGEBRA. 

5.  Differentiate  ^  =  r*  —  ic^,  and  explain  the  significance  of  the 
result  as  above.  Result,    dij  —  (5.r*  —  Zx^)dx, 

6.  In  order  to  keep  the  relation  "ly  —  3.c*  true  as  x  varies,  how- 
must  y  vary  in  relation  to  a;  ?  What  is  the  relative  rate  of  change 
when  a;  =  4  ?    When  a;  =  2  ?    When  a;  =  1  ?     When  a;  =  J  ?    When 

Anstvers.  When  a:  =  4,  y  increases  13  times  as  fast  as  x.  AVhen 
X  =  i,  y  increases  at  the  same  rate  as  x.  In  general  //  increases  3x 
times  as  fast  as  x.    Wlien  x  is  less  than  |,  y  increases  slower  than  x. 

2x^             X*  —  1 
7  to  12.  Differentiate    the    following:     u  =  -— ;    u  =  ^ ; 

3y  x^  -\-  I  ^ 

y  =  x^z^]       n  =  xhf  4-  6a;;       ^  =  .-c*  -  3a^  +  4a:»  -  a;'  +  1  ;       and 
2/  =  K  -  K  +  ^• 

13  to  17.  Differentiate  y=(«2  4-a;3)5  ;  ^=(a  +  a;«)*;  2^=(3a;-2)^; 
2/  =  (2  -  .r2)-2  ;  and  ?/  =  (1  +  x)~^. 

Sug's. — Such  examples  should  be  solved  by  considering  the  entire  quantity 

within  the  parenthesis  as  the  variable.    This  is  evidently  admissible,  since  any 

expression  which  contains  a  variable  is  variable  when  taken  as  a  whole.    Thus 

a. 
to  differentiate  y  —  {a  +  cc*)  ,  we  take  the  continued  product  of  the  exponent  {%), 

the  variable  {a  +  x^)  with  its  exponent  diminished  by  1,  [t.  e.,  {a  +  x^)~^],  and 
the  differential  of  the  variable  {i.  e.,  the  differential  of  a  +  x*,  which  is  'Zxdx). 

_i  _i  4xdx 

This  gives  us  dy  =  §(a  +  x')  ^2xdx,  or  dy  =  ^iX(a  +  x')  ^dx  =  — ^ 


'dVa 


+  x^ 


18  to  22.  Differentiate^ ;   ^^ r:; :   — -:  —m- 


i  +  x'  (1  +  xY'  (1  +xy'     "\i  -i-xY 

and  —  in  —— — r^ . 
(1  -f   xY 

23.  In  the  expression  Ga:^,  when  x  is  greater  than  1  does  the  func- 
tion (Ga;^)  change  faster  or  slower  than  x  ?  How,  when  ,r  is  less 
than  ^?    What  does  the  process  of  differentiating  eaj*  signify  'f 

Answer  to  the  Inst.  Finding  the  relative  rate  of  change  of  6.c '  and  x,  or  find- 
ing what  increment  6.r'  takes  when  x  takes  the  increment  dx. 

Or,  in  still  other  words,  finding  the  difference  between  two  consecutive  states 
of  6a;^  and  hence  the  relation  between  an  inflnitesimal  increment  of  x  and  the 
corresponding  increment  of  Qx\ 


INDETERMINATE   COEFFICIENTS.  159 


SECTION  IT. 

INDETERMINATE  COEFFICIENTS. 

158,  Indeterminate  Coefficients  are  coefficients  assumed 
in  the  demonstration  of  a  theorem  or  the  solution  of  a  problem, 
whose  values  are  not  known  at  the  outset,  but  are  to  be  determined 
by  subsequent  processes. 


159.  Prop— If  A  +  Bx  +  Cx'+  Dx^  +  etc.  =  A'  +  B'x  +  CV 
-f  D'x'+ e^c,  in  which  x  is  a  variable*  and  the  coefficients  A,  B, 
A',  B',  etc.  are  constants,  the  coefficients  of  the  like  poicers  of  x  are 
equal  to  each  other.  That  is,  A  =  A'  (these  being  the  coefficients  ofx^), 
B  =  B',  C  =  C,  etc. 

Dem.— Since  the  equation  is  true  for  any  value  of  w,  it  is  true  for  x=0.  Substi- 
tuting tliis  value,  we  liave^  =  ^'.  Now  as  A  and  A'  are  constant,  they  have  the 
same  values  whatever  the  value  assigned  to  x.  Hence  for  any  value  of  x,  A  — A'. 
Again,  dropping  A  and  A',  we  have  Bx  +  Cx^+  I>x^+  etc.  =  B'x  +  C'x^-j-  D'x'^ 
•f  etc.,  which  is  true  for  any  value  of  x.  Dividing  by  x,  Ave  obtain  B+C.v+Dx'' 
+  etc.=B'  +  C'x  +  D'x^+  etc.,  likewise  true  for  any  value  of  x.  Making  x  =  0, 
JB  =  ^',  as  before.  In  this  manner  we  may  proceed,  and  show  that  C=C 
B  =  D' ,  etc.    Q.  E.  D. 

160.  Qo^.—If  A  +  Bx  -I-  Cx'^  +  Dx'  +  etc.  -  0,  is  true  for  all 
values  of  x,  each  of  the  coefficients  A,  B,  0,  etc.^   is  0. 

For  we  may  write  A  +  Bx  +  Cx^  +  Bx^  +  Ex*  +  Fx^  ■+-  etc.  =  0  +  0.c  +  Ox- 
+  Ox^+  Ox*+  Oir'+  etc.     Whence  by  the  proposition  ^  =  0,  ^  =  0,  C=  0,  etc 


Development  of  Functions  by  means  of  Indeterminate 
Coefficients. 

101,  A  Function  is  said  to  be  Developed  when  the  indicated 
operations  are  performed;  or,  more  properlv,  when  it  is  transformed 
into  an  equivalent  series  of  terms  following  some  general  law. 

Ill's, — Division   affords   a  method  of  developing  some  forms  of  functions, 

♦  Saying  that  r  it  a  variable,  is  eqnivalent  to  saying  that  tlie  equation  must  be  true  for  any 
value  of  X.  Thii?  it?  an  cssontial  thing  in  this  discussion.  The  members  of  such  an  equation 
are  somsiimos  said  to  be  Identically  equal. 


160 


ADVANCED  COURSE  IN  ALGEBRA. 


Thus    y  = 


when  developed  by  division  becomes  y  =  \  +  x  +  x^  +  x^-^  etc. 


The  binomial  formula  (Complete  School  Algebra,  195,  or  168  of  this 
treatise)  is  a  formula  for  developing  a  binomial.  Thus  y  —  (a-\-  xf  when  devel- 
oped becomes  y  =  a^  -\-  ba\T -\-\Oa^x- -^-Kki^x^ -^ ^nx*  +  x^ .  The  subject  is  one 
of  great  importance  in  matliematics,  and  the  method  of  Indeterminate  Coeffi- 
cients forms  the  basis  of  most  that  is  valuable  upon  it. 


Examples. 


\-x 


1.  Develop r-  into  a  series  by  the  method  of  Indeterminate 

^  l-{-x-\-ar 

Coefficients. 

1-3 


Solution. — Assume 
of  fractions, 


\-^x-\-x^ 


A  +  Bx  +  Cx^-ir  i).c '  +  Ex^±  etc.    Clearing 


A+n 

■¥A 


+B 

^A 


x'+B 

+  C 


x'  +  E 
+B 

4-6' 


etc. 
etc. 
etc. 


Equating  the  coefficients  of  the  corresponding  powers  of  x  by  {l/>0),xve  have 
the  following  equations  from  Avhich  to  find  the  values  oi  A,  B,  C,'B,  etc. : 
A=l',  A+B=-l;  A-hB+C==0;  B-hC+B  =  0;  C-\-B+E=0.  Solving 
these,  we  have    ^  =  1,    B=—2,    C=l,    D=l,  and  i^=  —  3. 

Substituting  these  in  the  assumed  development,  we  have 

\—x 

^  =  l—2x+x^+x^~2x*+  etc. 

1+x  +  x* 

This  can  readily  be  verified  by  actual  division. 

2.  Develop,  or  expand  into  a  series  (a'—  x^y  by  means  of  Inde- 
terminate Coefficients. 

Solution. — Assume 

{a-  -  x^)^=  A+Bx+Cx* +Bx^-hEx* +Fx'^  -^Gx'^  +  etc. 
Squaring  both  members  and  expanding  {a^—x^y,  we  have 


a' 

-da'x'-+daKi*'-x'^A^+AB 

x+ACx''\^AD 

.v*+AEx*'{-AF,c'  +  AG 

A""  4- etc. 

+AB 

+-B*      '¥BC      -^BD      +BjJ     -\-BF 

-fete. 

+AC    -\BC 

+C'  i   -hcn    +CE 

+etc. 

\ 

■¥AD 

+BD\     +CD     -f/;* 

4- etc. 

■\-ABi\    -^BE     -^VE 

-l-etc. 

-^Ali',    +BF 

+etc. 

+A0 

+etc. 

Equating  the  cOefficiefits  of  the  cOri-espOndhig  powers  of  .t,  wc  find  A^  —  a^, 
OtA-  tt ' ;    2AB  s  0,  whence  ^  =:  0  ;    2^C  -h  -B^  =  -  3a^  whence  f  =  -  ^n  ; 

2Ui)  ^  SO)  «  0,  whence  2)  =  0 


2(AS  -h  BB)  +  C^  =  3a ^  .  whence  E^^\ 


,  indeter:minate   coefficients.  161 

2[AF-irBE-\-  CD)=0,  whence  F—0 ;  and  in  like  manner    G  = -„  etc.    (If  the 

feipansion  of  the  second  member  had  hern  carried  farther,  each  of  the  succeeding  co- 
efficients would  be  equated  with  0,  as  there  are  no  terms  in  the  first  member  contain- 
ing higher  powers  of  a;  than  the  Ctli.)  Substitutingthe  values  of  y1,  B,  C,  jD^  etc,  as 

now  found,  we  have     (a^—x^)'=a^—  ^ax-+  -^ — f-  v;r-5  +  etc. 

3.  Expand,  or  develop  (1  —x^)^  by  meciiis  of  Indetenniuate  Coeffi- 
cients.     Also  — -,     .; ,    and 


x  +  V    b-ax'  (i_.c)i* 

SuG. — To  expand  the  last,  put  the  expression  equal  to  the  usual  series, square 
both  members,  and  then  clear  of  fractions. 


102,  ScH. — In  using  the  method  of  Indeterminate  Coefficients,  as  the 
series  A  +  Bx+  Cx' -\-  etc.,  is  merely  hypothetical  at  the  outset,  we  must 
carefully  observe  whether  the  subsequent  processes  develop  any  inconsist- 
ency. For  example,  perhaps  a  particular  expression  will  not  develop  in  the 
form  assumed.     If  so,  some  inconsistency  will  appear  in  the  process.     Thus, 

2                                        2 
were  we  to  attempt  to  develop  -j~ — -^  by  assuming   ~, .^  —  A  -\-  Bx  -}-  Cx'^ 

+  Dx'*+  etc.,  we  should  find,  after  clearing  of  fractions,  that  the  first  mem- 
ber had  only  tl:e  term  2,  which  is  2x" ;  and  as  there  would  be  no  correspond- 
ing term  in  the  second  member,  we  should  have  to  write  2  =  0,  which  is 
absurd.  In  general,  we  observe  that,  when  we  equate  the  coefficients,  the 
second,  or  assumed  member,  must  have  a  term  containing  as  low  a  power  of 
the  variable  as  the  lowest  in  the  first  member.  This  may  be  secured  either 
by  putting  the  expression  to  be  developed  into  a  proper  form  before  assum- 
ing the  scries,  or  by  assuming  a  series  of  proper  form.     Thus,  in  the  above 

2  12  2 

case,    we  may   write   for  — ,    — -  • ,    and   then   develop by 

x^  —  x^x*l—x  1—x 

2 
assuming    ■ --  =  A  +  Bx  +  Cx^+Bx"^  +  etc.,    and   finally   multiplying  by 

1  2 

—  ;  or  it  may  be  developed  by  assuming  — ^  =  Ax~^+Bx-^+Cx^+  Bx 

-f  Ex^  -+-  etc. 


lT    —    "ix    •+-   Z 

4.  Expand  — , — r-  by  the  method  of  Indeterminate  CoefR- 

'X  — ~  iC  ~r   X 

.-        1-1- 2a:  .,  l-x 

cients.     Also ^r^. .      Also 


3:r*  2x'  +   ^x^ 


5.  Expand   y"!  -x..     Also  (1  +  x)^. 

11 


162  ADVANCED  COUKSE  IN  ALGEBKA. 


Decomposition    of   Fractions   by    means    of    Indeterminate 

Coefficients. 

163.  For  certain  purposes,  especially  in  the  Integral  Calculus, 
it  is  often  necessary  to  decompose  a  fraction  into  partial  fractions. 
There  are  three  principal  cases. 

104:,  Case  1. — A  fraction  which  U  a  function  of  a  single  varia- 
ble^ whose  numerator  is  of  lower  dimensions  than  its  denominator, 
and  whose  denominator  is  resolvable  into  n  REAL  and  unequal  fac- 
toi's  of  the  first  degree^  can  be  decomposed  into  n  partial  fractions  of 

,     ^           A              B             C                                    N 
the  form 1 r  ^ • >  x  +  a,  x  +  b, 

''         x  +  ax  +  bx  +  c  x+n 

X  +  c,    -     -     -     -    -     x+n  being  the  factors  of  the  denominator. 

M*         ABC                              N         . 
Dem.— Assume     ---^    = +    r  + ,     in 

q){u-)         X  +  a        X  +  b        x  +  c  x  +  n 

which  f{x)  is  of  lower  dimensions  f  than  q){z),  and  <p{x)  =  {x  +  a){x  +  b)  {x  +  c) 

(x  +  n).  I     Reducing  the  partial  fractions ,     r  ,  etc.,  to 

forms  having  a  common  denominator,  this  denominator  will  be  the  product  of  all 
the  denominators  x  +  a,  x  +  b,  x  +  c,  etc.,  and  hence  will  be  (p{x),  and  each 
numerator  will  contain  one  less  of  these  factors  than  the  common  denominator, 
and  hence  will  be  of  the  {n  —  l)th  degree,  the  denominator  being  of  the  wth  de- 
gree.g  Then,  as  the  denominators  of  both  members  will  be  equal,  the  numera- 
tors will  also  be  equal.  Placing  them  so,  we  can  find  the  values  of  the  indeter- 
minate coefficients  A,  By  C,  etc.,  by  the  principle  in  {139).  The  necessity  for 
having /(a-)  of  lower  dimensions  than  (p{x)  is  the  same  as  is  pointed  out  in  {102). 
Thus,  if  j\x)  contained  a  term  like  5^;*  while  (p{x)  contained  none  higher  than 
2a;*,  we  should  be  required  to  write  5  =  0,  as  there  would  be  no  term  in  the  sec- 
ond member  having  an  «'  in  it.    Finally,  having  obtained  the  values  of  A,  B,  C, 

ABC 

etc.,  we  can  substitute  them  in   ,      ,      ,     etc.,  and   have   the 

X  +  a       X  +  b       X  +  c 
partial  fractions  sought. 


lOS,  Case  2. — A  fraction  which  is  a  function  of  a  si?igle  varia^ 
ble,  whose  numerator  is  of  lower  dime?isions  tha?t  its  de?iominator, 
and  whose  denominator  is  resolvable  into  n  real  a)id  TiQUALfactoi's 

*  See  (139,  140). 

t  That  i:?,  does  not  contain  so  high  a  power  of  x. 

X  The  proposition  aspumes  that  cp{x)  is  resolvable  into  n  real  and  unequal  factors  of  the 
fir<t  degree. 

§  That  is,  containing  x  to  the  nth  power,  and  no  higher  power. 


PBCOMPOSITION   OF  FRACTIONS.  163 

of  the  first  degree^  can  be  decomposed  into  ii  partial  fractions  of  the 
.  A  ]^  C  N 

form     J—- r-,     +    — — r-3y     + 


(x  +  a)'        (X  +  a  y-'        (x  +  a)"-=^  x  +  a' 

X  +  a  being  one  of  the  equal  factors  of  the  denominator. 

Dem.— Assume   ^,  =  ; ^+^ — ; — ;— f  4  7- ^ — ;     -     -     -     -      -=^— , 

in  which  J\.v)  is  of  lower  dimensions  than  q){x),  and  (p[x)  =  {x  +  a)"".  Reducing 
the  partial  fractions  to  forms  having  the  common  denominator  {x  +  a)"  (i.  e.  <p(,x)), 
and  placing  the  numerators  of  the  members  equal,  we  Lnd  tiiat  thu  tecoiid  mem- 
ber is  not  of  lower  dimensions  with  respect  to  the  variable  x,  than  the  first  mem- 

N 
ber,  since  tlie  niiinerator  of  the  fraction  -; will  contain  the  highest  power  of 

X  of  any  of  the  terms,  and  this  will  have  no  higher  power  than  2?"-^  as  (a? +  «)*"* 
is  the  factor  by  which  the  terms  of  the  fraction will  be  multiplied  in  the  re- 
duction. Hence,  we  can  find  the  values  of  A,  B,  C,  etc.,  by  {15i)),  and  these 
substituted  in  the  assumed  series  will  give  the  required  partial  fractions. 


100,  Case  3. — A  fraction  which  is  a  functioii  of  a  single  varia- 
ble, whose  numerator  is  of  lower  ditnensions  than  its  denominator, 
and  whose  denominator  is  resolvable  into  n  real  and  equal  quad- 
ratic yac^ors,  can  be  decomposed  into  n  vartial  fractions  of  the  form 
Ax  +  B  Cx  +  D  Ex  +  F 


[(x  +  a)"  +   b'^]-       '       L(x  +  ^)'  -r  V]"-'  [(X  +  a)^  +  b^t 

Mx  +  N 


(X  +  a)'  +  b^ 
factors  of  the  denominator. 


(x  +  a)''  +  b'  being  one  of  the  equal 


Dem. — Assume 

f{x)  _  Ax  +  B Cx  +  B  ^   Ex  +  F 

^)  "   [{x  +  ay  +  6=^]"    "*"    [(3  +  ay  +  &'^]"-»  ^    [:-  -r  af  +  Z»=^]»-« 

Mx  +  N 
{X  +  a)*  +  6* ' 

Bringing  the  terms  of  the  secoBid  member  to  a  common  denominator  q){x),  or 
[{x  -f  ay  +  &*]",  we  find  that  the  highest  power  of  x  involved  in  the  numerators 
is  «*"-',  which  will  ari.e  in  multiplying  Mx  -{•  N\)y  \{x  +  ay  +  ft^]"-'.  But,  as 
f{x)  IS  of  lower  dimensions  than  cp{x),  and  (p{x)  is  of  2n  dimensions,  the  numera- 
tor of  the  second  member  will  not  be  of  lower  dimensions  than  j\x),  and  hence 
equating  theni,  the  values  of  A,  B,  G,  etc.,  can  be  determined  and  substituted  in 
the  assumed  series  of  parti?,!  fractions. 

107.  Sen.— When  the  dcnomir.ator  of  the  fraction  to  be  decomposed  is 
composed  of  factors  of  two  or  more  of  the  forms  referred  to  in  the  three  given 


164 


ADVANCED  COURSE  IN  ALGEBRA. 


cases,  the  forms  of  the  assumed  partial  fractions  must  be  made  to  correspond. 
Thus  were  it  required  to  decompose  ^j._i,^^^^ayix' +a-')* '  *^®  assumed  par- 

tial  fractions  would  be  —  + -,  +  ; r^  +  7 rr  + +     ^         ,-; 

X      x  —  b      {X  +  ay      {X  +  ay      x  +  a      {X*  +  a*y 

Hx  +  I 

'^  x*+a*' 


Examples. 
1.  Decompose 3  into  partial  fractions. 

»  ^  x*  -2     A        B  C  ,  ^.v. 

Solution.— Assume =  —  + + ,  x,l  —  x,  and  1  +  a;  bemg 

x  —  x*xl  —  xl  +  x  ^ 

the  unequal  factors  of  2;  —  x'^  {115,  117).  Bringing  the  terms  of  the  second 
member  to  a  common  denominator,  we  have 

x'  -2  _A-  Ax*  +  Bx  +  Bx*  +  Cx  -  Cx* 
x-x^~  x{X  —  x)(X  +  x) 

Hence  x*  —  2  —  A  ^■  {B  -^t  C)x  ^  {B  —  A  —  C)x*  ;  from  which  we  get  ^  =  -  2, 
B  +  C  =  0,  and  B  —  A  —  C  =1.  Solving  these  equations  we  find  A  =  —  2, 
B  =  —  ^,  and  C  =  i.    These  values  inserted  in  the  assumed  forms  give 

«^--2  _  -2 i_      ^_  ^  _  2 1 1 

x^x^~    X        i-a:'^l+a;  x      2(1  -  i-)  "^  2(1  + a;)* 

«i«T^                      .i*.ii-                    X  -{■  d                   X  -{•  1 
2  to  6.  Decompose    the   following :      -, ;        —, -rr  ; 

X^   —   X   —   -v  XKX  ~'~  /C) 

a;  +  1  3a;  —  5  ,  x* 

-^;    and 


x"  -  Ix  +  12'    a?  -  Qx  +  %'  a;'  +  6a;'  +  11a:  +  6  * 

Suo. — In  case  the  factors  of  the  denominator  are  not  readily  discerned,  place 
the  denominator  equal  to  0  and  resolve  the  equation.  Thus  the  last  example 
^ves  «'  +  6x*  4-  11a;  +6  =  0.  From  which  we  have  a;  =  —  1,  —  2,  and  —  3 
{119),  and  the  factors  are  a;  +  1,  a;  +  2,  and  a;  +  3. 

7  to  11.  Decompose  the  fractions  -(^3^)3-  5  ^^^x^+^zu  +  2r 

and 


a:^(l  -  a;')(l  +  a;) '    a;*  -  1'  (x  -  2)>  4-  3)'-'* 

a:*— 2a; +  3      3ar^—    a^- 10.^*+ 15a;»4- 2a;  — 8 


12  to  18.  Decompose 


(a;«  +  l)'    '  a-(a;^-2)'(a;-l) 


a;'-a;  +  l         1  1  1  ^       6a;2-4a;-6 

and 


x'ix  +  1)'  ^-  1'  a'-a;*'  x'-  (a+b)x-{-ay  ar^- 6a;H  lla:-6 


THE   BIKOMIAL  FOKMULA.  165 


SECTION  IIL 

THE  BINOMIAL   FORMULA. 

108,  Theove^n, — Letting  x   and  y  repi'esent   any  quantitiea 
whatever  (i.  e.  he  variables)  and  m  any  constant, 

Dem. — We  may  write  (r+y)"*  =  a;"*  (  1  +  -  )   .     Now  put  '-   =  s  and  assume 

(1  +  2)»  =  .4  +  Z?z  +  Cfe'  +  Dz^  +  ^2^  +  Fs**  +  etc.,  (1) 

in  which  A,  B,  C,  etc.,  are  indeterminate  coefficients  independent  of  z  {i.  e.  con- 
stants), and  are  to  be  determined.  To  determine  these  coefficients  we  proceed 
as  follows  : 

Differentiating  (1),  we  have 

m{\  +z)'''-^dz=Bdz+2Czdz-\-SDz'^dz+iEz^dz+5Fz*(k-\-  etc. 

Dividing  by  dz,  we  have 

m(l+zr-'=B+2Cz  +  3i)2=  +  4Ez'+5FY+  etc.  (2) 

Differentiating  (2)  and  dividing  by  dz,  we  have 

mim-l){l+zr''=2C+  2  -  32)2  +  3  •  AEz'-\-  4  -  5i^='4-  etc.  (3) 

Differentiating  (3)  and  dividing  by  dz,  we  have 

7n(7/i-l)(m-2Xl+2)'"-'=2  ■  32)  +  2  •  3  •  4£fe  +  3  -  4  -  5^''  +  etc.      (4) 

Differentiating  (4)  and  dividing  by  dz,  we  have 

m(m-lXw-2)(w-3Xl+2)"-^=2  •  3  -  4^  +  2  -  3  -  4   Si^^s  +  etc.        (5) 

Differentiating  (5)  and  dividing  by  dz,  we  have 

w(m-lXw-2Xm-3Xm-4Xl+2)'"-'=2  -  3  ■  4  •  5F+  etc.  (0) 

We  have  now  gone  far  enough  to  enable  us  to  determine  the  coefficients  A, 
B,  C,  D,  E,  and  F,  and  doubtless  to  determine  the  law  of  the  series. 

As  all  the  above  equations  are  to  be  true  for  all  values  of  z,  and  as  the  coeffi- 


*  This  form  is  read  "  factorial  3,"  "  factorial  4,"  etc.  ;  and  eignifiee  the  product  of  tlic  nat- 
ural numbers  from  1  to  3, 1  to  4,  etc. 


100  ADVANCED  COURSE  IN  ALGEBRA. 

cients  A,Ii,  C,  etc.,  are  constants,/,  e.,  have  the  same  values  for  one  value  of  z 
as  for  another,  if  we  can  determine  their  values  for  one  value  of  z,  these  will 
be  their  values  in  all  cases.     Now,  making  2=0,  we  have  from  (1)  A  =  l  ;  from 

(2),  B  =  711  ;  from  (3),  C  = — (the  factor  1  being  introduced  into  the  de- 

nominator  for  the  sake  of  symmetry) ;  from  (4),  D  =  -p ;  from  (5), 

11 

IL  = p ;  from  (6),  F=  ^ . 

These  values  substituted  in  (1)  give 

m{m  —  1) ,      w(m  -  \){m  —  2)  ,     m(m— l)(m— 2)(m— 3)  ^ 

-j2— ^   +  [3  '  +  ji  ' 

yn(m-l)(7n-2)(7n-3)(7n-4)_, 
-\  j^j  z  -+-  eic. 

Finally,  replacing  e  by  its  value  -,  we  have 

m{m  —  \)(m  -  2){m  —  3)  y*        m(,m  -  \\m  —  2)(m  —  3)(m  -  4)  y'  ) 

-*■    [4 ^^  + j5^ ^   +•^^^•1 

=af'  +rnx-'2,+  ^^^i)r"-y+  m(m-lXm-2)^.3^_^m  (m-1)  (m-2)(m-8) 

LI  I?.  li 

^  ^      rw(m-lKm-2Xwi-3)(m-4) 
af-V  +  -^^ ^ -^aJ— V  +  etc. 

109,  Cor.  1. —  Tlte  nth,  or  general  term  of  the  series  is 
m(m  —  1)  (//;  —  2) {in  —  n  +  2)  ^,_ 

\n-l 


af'-*+^y*-\ 


For  we  observe  that  the  last  factor  in  the  numerator  of  the  coefficient  of  any 
particular  term  is  m  —  the  number  of  the  term  less  2,  i.  e.,  for  the  nth  term, 
m  —  {n—2),  or  771  —  n  +  2  ;  and  the  last  factor  in  the  denominator  is  the  number 
of  the  term  —  1,  i.  e.,  for  the  r<th  term,  n  —  1.  The  exponent  of  x  in  any  par- 
ticular term  is  m  —  the  number  of  the  term  less  1,  i.  e.,  for  the  wth  term, 
7n—{n  —  1),  or  7n  —  n  4-  1  ;  and  the  exponent  of  y  in  any  term  is  one  less  than 
the  number  of  the  term,  i.  e.,  for  the  nth  term,  n  —  \. 

170,  Def. — In  a  series  the  Scale  of  delation  is  the  relation 
which  exists  between  any  term  or  set  of  terms  and  the  next  term  or 
set  of  terms. 

171,  CoR.  2. —  The   scale   of  relation  in  the  binomial  series  is 

I —  1  )-»  since  the  nth  term  multijMed  hy  this  produces  the 

(n  +  1  )th  term. 


TKi:    BINOMIAL   FORMULA.  167 

This  is  readily  seen  by  inspecting  the  series,  or  by  writing  the  (a  +  l)th  term 

and  dividing  it  by  the  nth.     'J'iius,  substituting    in  the  general   term  as  given 

.    .  ,  m(w— l)(m— 2) (m—n+1) 

above,  n  +  1  for  n,  we  have    ; ^^ -V'-"  v»,  as  the 

(rt  +  l)th  term.     This  divided  by  the  nth,  or  preceding  term,*  gives  —  , 

Examples. 

1  to  6.  Expand  the  following  :    (a  —  b)^ ;    (x  —  y)'^ ;    (a  —  a;)"* ; 
{l-\-xy;    {l-yy;    (1  -  y)". 

7  to  11.  Expand   {x  +  y)-' ;     (x  -  y)-' ;     {a  -  x)-' ;     ^^  ^  ^^,  ; 
,    or    {x-\-  y)-\ 


x  +  y 


[Note. — For  practical  suggestions  in  the  use  of  this  theorem,  see  Complete 
School  Algebra,  pages  148-154,  or  Part  I.  of  this  volume,  pages  58,  59.] 

12.  Expand  {a  +  xy  by  using  the  scale  of  relation. 

Solution. — The   scale  of    relation    { 1  )-     becomes    in  this     case 

V    n  Jx 

( 1  )-  .      Now  the  first  term  is  a^ .     To  obtain  the  next  ti  =  1,  whence 

\    n  J  a 

the  scale  of  relation  5  -  .      Multiplying  a'  by  this  scale  of  relation,  we  find  the 

a 

X 

second  term  5a*x.     For  the  next  the  scale  of  relation  is  2  - .      Hence   tlie  3d 

a 

term  is  lOa'a;*.     For  the  next  the  scale  of  relation  is  -,     giving  for  the  4th 

a 

(6        \x  X 

T  —  1  )-    or  A—  , 
4        /a  a 

giving  for   this  term  5ax*.      For  the   Gth   term  the  scale  of  relation  equals 

/6         \  X  X  /6         \  X 

I  -  —  1  I—  or  i-,  giving  x^.    For  the  7th  term  the  scale  of  relation  is  (  ^  —  1 )  - 

or  0.     Hence  the  series  terminates, 

13.  Expand  {m  —  n)~^  by  nsing  the  scale  of  relation,  and  also 
by  the  general  formula. 

14  to  17.  Expand  (1  -  a^)^ ;     (3  +  x^)^;  {x  -  y)'^ ;      (a  +  x)'^. 

*  The  numerator  of  the  coefficient  of  the  preceding,  ornth  term,  contains  all  the  factors  of 
the  numerator  of  the  (rt  +  l)th  except  m  -  n+1,  as  the  factor  in  the  (n+l)th  preceding  m  -  n  +  1 
is  m  -  n  +  2,  etc.    Similarly  in  the  denominator. 


168  ADVANCED  COURSE  IN  ALGEBRA. 

18  to  20.  Expand  (a^-x^-)^  ;     (U  -  x^)-' ;      (a^  +  c*)*. 

SuG's.— In  cases  in  which  the  terms  of  the  binomial  are  not  single  letters  or 
figures,  it  will  be  best  to  substitute  single  letters,  expand  and  then  replace  the 

values.  Thus,  to  expand  (a; '  — 3«)"='", put  x-^=  y,  and  3rt  =  h,  and  expand  (y— 5)"^; 
and  in  this  expansion  restore  the  values  of  y  and  h.  In  like  manner  the  for- 
mula may  be  applied  to  any  polynomial.  Thus,  to  expand  (1  —  ar*4-  3y)',  put 
(1  —  «*)  =  e,  and  3y  =  u,  expand  («  +  «)*,  and  then  restore  the  values. 

21.  Expand  -  into  a  series. 

SuG'8.        .     ^ =a(6'-c^Jr')~^     Put  6*= -p,  and  c»^«=  y,  and  expand 

(«  —  y)~*,  etc.     The  result  is 

,c»a^     1    3     cV    13    5     c^^•    1  -3-5-7     cV  ^ 

&»"^2.4"    i.*"^2    4-6"    i."  "^2-4    6-8  ■  "fe^"^  ®*^*f 

22.  What  is   the  4th  term   of  the   development  of    (a^+z)*? 
(See  169>i 

SuG.-The  general  term  is  ^^  -  D (m-n  +  1)  ^.„_.+,  ,_,     i^  ^his 

[n  — 1 

case    m  =  i,    n.  =  4,    a;  =  a*,    y  —  z.     Whence  the  4th  term  is  — 

16a' 

23.  What  is  the  7th  term  of  (a'^-h^)^'t    The  10th  term? 


SECTION   IV, 

LOGARITHMS. 

^7^.  A  Logarithm  is  the  exponent  by  which  a  fixed  number 
is  to  be  affected  in  order  to  produce  any  required  number.  The 
fixed  number  is  called  the  Base  of  the  System. 

III.— Let  the  Bate  be  3:  then  the  logarithm  of  9  is  2 ;  of  27,  3 ;  of  81,  4 ; 

of  19683,  9 ;  for  3*=  9  ;  3=^=  27  ;  3*=  81  ;  and  3^=  19683.     Again,  if  64  is  the 

base,  the  logarithm  of  8  is  i,  or  .5,  since  64%  or  64''=  8;  i.e.,  |,  or  .5  is  the 

exponent  by  which  64,  the  base,  is  to  be  affected  in  order  to  produce  the  num- 

j. 
ber  8,     So,  also,  64  being  the  base,  i,  or  .333+  is  the  logarithm  of  4,  since   64^, 


LOGARITHMS.  169 

or  04'"^—  4  ;  i.  e.,  }i,  or  .333+  is  the  exponent  by  which  64,  the  base,  is  to  be 

affected  in  order  to  produce  the  number  4.   Once  more,  since  64^,  or  64*^^^+=16, 

§,  or  .666-1-  is  the  logarithm  of  16,  if  the  base  is  64.  Finally,  64~^  or  64—^ 
=  -},  or  .125  ;  hence  — ^,  or  —.5  is  the  logarithm  of  \,  or  .125,  when  the  base  is 
64.  In  like  manner,  with  the  same  base,  — ^,  or  —.333+  is  the  logarithm  of  \, 
or   .25. 

173,  COH. — Since  any  number  icith  0  for  its  €xpo7ient  is  1,  the 
logarithm  of  1  is  0,  in  all  systems.  Thus  10^=  1,  ichence  0  in  the 
logarithm  of  1,  ^V^  a  system  in  \chich  the  base  is  10. 

174,  A  System  of  Lof/aritluns  is  a  scheme  by  which  all 
numbers  can  be  represented,  either  exactly  or  approximately,  by 
exponents  by  which  a  fixed  number  (the  base)  can  be  affected. 

175,  There  are  7\vo  Systems  of  Logaritlims  in  common  use, 
called,  respectively,  the  Briggean  or  Common  System,  and  the  Na- 
pierian or  Hyperbolic  System.*  The  base  of  the  former  is  10,  and  of 
the  latter  2.71828 +.  In  the  present  treatise  we  shall  confine  our 
attention  to  systems  whose  bases  are  greater  than  1. 

176,  Cor.  1. — Neither  1  7ior  any  iiegative  number  can  he  used 
as  the  base  of  a  system  of  logarithms. 

For  all  numbers  cannot  be  represented  either  exactly  or  approximately  by  ex- 
ponents of  such  numbers.  Thus  with  1  as  a  base  we  can  represent  no  other 
number  than  1  by  its  exponents,  for  1  with  rt?i</  exponent  is  1.  Moreover,  with  a 
negative  base  the  logarithms  which  were  odd  numbers  would  represent  negative 
numbers,  and  those  which  were  even  numbers  would  represent  positive  numbers. 
For  example,  with  —2  as  a  base,  3  might  be  considered  as  the  logarithm  of  —8, 
since  (— 2)  '=  -  8  ;  but  no  number  could  be  found  as  a  logarithm  to  correspond 
to  8  (*.  e.  +8),  since  —2  cannot  be  affected  with  any  exponent  which  will  pro- 
duce 8. 

177 ,  One  of  the  most  important  uses  of  logarithms  is  to  facilitate 
the  multiplication,  division,  involution,  and  the  extraction  of  roots 
of  large  numbers.  These  processes  are  performed  upon  the  following 
principles : 

17 H,  I^rop.  1, —  The  logarithm  of  the  product  of  two  7iumbers 
is  the  sum  of  their  logarithms, 

Dem. — Let  a  be  the  base  of  the  system.  Let  m  and  n  be  any  two  numbers 
whose  logarithms  are  x  and  y  respectively.    Then  by  definition  a'^m,  and  00=71. 

*  The  common  syBtem  is  the  one  used  for  practical  p^rRoges.  a»d  the  only  one  of  which 
there  are  table?  in  com -non  use,  Napierian  Ipgarithnis  j^re  HStially  in^plied  in  abstj-act  mathe- 
matical difciipgion. 


170  ADVANCKI)    COLKSE    l.\    ALGKBRA. 

Multiplying  tUo  corresponding  members  of  these  equations  together  we  Lave 
«'  '—ma.     Whence  x  +  y  \»  the  logarith  of  mn.    Q.  e.  d. 

170,  J*l'op,  2* — The  logarithm  of  the  quotient  of  two  jrumbers 
is  the  lof/arithm  of  the  dividend  minus  the  logarithm  of  the  divisor. 

Dem. — Let  a  be  the  base  of  the  system,  and  m  and  n  any  two  numbers  whose 
logarithms  are,  respectively,  ot,  and  y.     Then  by  definition   we  have   a'=zm, 

and  W  —  n.      Dividing,  we  have   a""-"  =  — .     Whence  x  —  yia  the  logarithm 

-  m 
of  — .     Q.  E.  D. 

ISO,  Prop,  3, —  T7ie  logarithm  of  a  power  of  a  number  is  the 
logarithm  of  the  number  nndtiplied  by  the  index  of  the  j^ower. 

Dem. — Let  a  be  the  base,  and  x  the  logarithm  of  m.  Then  n^=m ;  and  raising 
both  to  any  power,  as  the  2tli,  we  have  a*-=m-.  Whence  xz  is  the  logarithm  of 
the  2th  power  of  m.     Q.  E.  D 

IHl,  Prop,  4, —  77ie  logarithm  of  any  root  of  a  number  is  the 
logarithm  of  the  number  divided  by  the  number  expressing  the  degree 
of  the  root, 

Dem. — Let  a  be  the  base,  and  x  the  logarithm  of  m.    Then  a'=m.    Ex- 

X 

tracting  the  2th  root  we  have  a==  /^/m.     Whence  -  is  the  logarithm  of  -y^m. 

z 

Q.  E.  D. 

182,  It  is  evident  that  in  any  system,  the  logarithms  of  most 
numbers  will  not  be  expressed  in  integers.  Thus  in  the  common 
system  the  logarithm  of  100  is  2,  and  of  1000  3  ;  hence  the  loga- 
rithm of  any  number  between  100  and  1000  is  between  2  and  3,  i.  e. 
2  and  some  fraction.  This  fraction  is  usually  written  as  a  decimal 
fraction,  and,  as  we  shall  see  more  clearly  hereafter,  can  in  general  be 
expressed  only  approximately. 

ISS,  Tlie  Integral  Part  of  a  logarithm  is  called  the  Character^ 
istiCf  and  the  decimal  part  the  Mantissa, 

184,  JProp. — TTie  Mantissa  of  the  logarithm  of  a  decimal  frac- 
tioriy  or  of  a  mixed  number^  is  the  same  flw  the  mantissa  of  the  num- 
ber considered  as  integral^ 

*  Usaally,  in  speakinj?  of  logarithms,  if  no  particular  syetem  is  mentioned,  tlie  common 
system  is  to  be  understood  as  meant,  especially  when  practical  numerical  operations  are 
referred  to. 


JQ4.4641  86  

28456.73, 

^Q3>454l  86 

2845.673, 

102.454185  _ 

384.5673, 

101.464185  _ 

38.45673, 

JO**-***'  85  __ 

3.845673, 

384567.3 

=  5.454185, 

38456.73 

=  4.454185, 

3845.673 

=  3.454185, 

384.5673 

=  3.454185, 

38.45673 

=  1.454185, 

LOGAHITHMS.  171 

Dem.— It   will  be   found  hereafter  that  log  3845673=6.454185.     Now  this 
means  that  10^ •  ^  ^  4 1 8  s  -2845673.    Dividing  by  10  successively  we  have 
10«-45  4i85  ^  284567.3,  or  log 

or  log 

or  log 

or  log 

or  log 

or  log  3.845673  =  0.454185. 

Now  if  we  continue  the  operation  of  division,  only  writing  0.454185  —  1, 
1.454185,  meaning  by  this  that  the  characteristic  is  negative  and  the  mantissa 
positive,  and  the  subtraction  not  performed,  we  have 

IOT.45  4  1 85  _  2845673,  or  log  .3845673      =1.454185, 

10^.454  186  _  02845673,  or  log  .03845673     =3.454185, 

107.46  4  185  ^  003845672,  or  log  .003845673  =  3.454185, 
etc.,  etc.     Q.  E.  D. 

185.  Cor.  1. —  TTie  characteristic  of  the  logarithm  of  an  integral 
number,  or  of  a  mixed  integral  and  decimal  fractional  yiumher^  is  one 
less  than  the  number  of  integral  places  in  the  number. 

The  characteristic  of  the  logarithm  of  a  number  entirely  decimal 
fractional  is  negative  and  numericaUy  one  greater  than  the  number 
of  O'j  immediately  following  the  decimcd  point. 

Thus  the  characteristic  of  the  logarithm  of  any  number  between  1  and  10 
is  0,  between  10  and  100  1,  between  100  and  1000  3,  etc.  Or  let  it  be  asked, 
"  What  is  the  characteristic  of  the  logarithm  of  5136?  "  Now  this  number  lies 
between  1000  and  10000,  hence  its  logarithm  lies  between  3  and  4,  and  is,  there- 
fore, 3  and  some  fraction. 

Again,  as  to  the  numerical  value  of  the  characteristic  of  the  logarithm  of  a 
number  wholly  decimal  fractional,  consider  that  10~*  =  /j=.l ;  10~2=yiij=.01  ; 
10-3  =  j-ij-  =  .001.  Thus  it  appears  that  any  number  between  1  and  .1,  i.  e.,  any 
number  expressed  by  a  decimal  fraction  having  a  significant  figure  in  tenth's 
place,  as  .3564,  .846,  .1305,  etc.,  will  have  its  logarithm  between  0  (the  logarithm 
of  1)  and  —1  (the  logarithm  of  .1),  Hence  such  a  logarithm  will  be  —1  4-  some 
fraction  (the  mantissa).  In  like  manner,  any  number  between  .1  and  .01.  *.  e., 
any  decimal  fraction  whose  first  significant  figure  is  in  lOOth's  place,  as  .03568, 
.0956,  .01303,  etc.,  will  have  for  its  logarithm  —3  +  some  fraction. 

186,  Cor.  3. — The  common  logarithm  ofO  is  —  oo. 

Since  a  number  less  than  unity  has  a  negative  characteristic,  and  this  char- 
acteristic increases  numerically  as  the  number  decreases,  when  the  number 
decreases  to  0,  the  logarithm  increases  numerically  to  oo.  Hence  log  0=— oo. 
To  illustrate,  log  .1=1,  log  .01=  3,  log  .001  =  3,  log  .0001=4.  Hence  when  the 
number  of  O's  becomes  infinite,  and  the  number  therefore  0,  we  have  log  0 

=  —00.  •■ 


172  ADVANCED  COUIISE  IN  ALGEBRA. 


Computation  of  Logarithms. 

187 »  Tlie  Modulus  of  a  system  of  logarithms  is  a  constant 
factor  which  depends  upon  the  base  of  the  system  and  characterizes 
the  system. 

188 »  IProp* —  The  differential  of  the  logarithm  of  a  numher  is 
the  differential  of  the  number  mtdtiplied  by  the  modulus  of  the  system, 
divided  by  the  number  ; 

Or,  in  the  Napierian  system,  the  modulus  being  1,  the  differential 
of  the  logarithm  of  a  number  is  the  differential  of  the  number  divided 
by  the  number. 

Dem. — Let  X  represent  any  number,  i.  e.  be  a  variable,  and  n  be  a  constant 
such  that  y=oe*.  Then  log  y=n  log  x  {180).  Differentiating  y=x'*,  we  have 
dy=nx'*~^dx;  whence 

dy 

n=-^—=  ^  =  ^=JL  /IN 

^*"'^^     "^dx     y-dx      '^' 

XXX 

Again,  whatever  the  differentials  of  log  y  and  log  x  are,  n  being  a  constant 
factor,  we  shall  have  the  differential  of  log  y  equal  to  n  times  the  differential  of 
log  X,  which  may  be  written 

rfdog  y)=/i  .  ff(log  X),  whence  n  =  |}^.  (3) 

Now  equating  the  values  of  n  as  represented  in  (1)  and  (2),  we  have  ";    ^  ' 
dy 
=  -~.    Whence  d{log  y)  bears  the  same  ratio  to  — ,  as  d(log  x)  does  to  — .    Let 

X 

ni  be  this  ratio.     Then  <f(log  y)—  — -,  and  <^(log  x)— . 

y  ^ 

We  are  now  to  show  that  m  is  constant  and  depends  on  the  base  of  the 
system. 

To  do  this,  take  y—Z'"',  from  which  we   can  find  as    above  ?i'=r    ,^,       ' 

d(log  2) 

dy 

=-— — .    Now  as  m  is  the  ratio  of  <?(log  y)  to  — ,  it  is  also  the  ratio  of  (Z(log  z)  to 

z 

—  ;   and  <?(log  2)= .    Thus  we  see  that  in  any  case  the  same  ratio  exists  be- 

z  z 

tween  the  differential  of  the  logarithm  of  a  number  and  the  differential  of  the 

number  divided  by  the  number.     Therefore  m  is  a  constant  factor. 


THE  LOGARITHMIC   SERIES.  173 

That  m  depends  upon  the  base  of  the  system  is  evident,  since  in  a  system  of 
logarithms  the  only  quantities  involved  are  the  number,  its  logarithm,  and  the 
base.  Of  these  the  two  former  are  variables ;  whence,  as  the  base  is  the  only 
constant  involved  in  the  scheme,  m  is  a  function  of  the  base.* 


189,  JProb. —  2'o  produce  the  logarithmic  series. 

Solution.— The  logarithmic  series,  which  is  the  foundation  of  the  usual 
method  of  computing  logarithms,  and  of  much  of  the  theory  of  logarithms,  ia 
the  development  of  log  (1  +  x).    To  develop  log  (1  +  x),  assume 

log  {1  i-x)  =  A  +  Bx+  Cx^  +  Bx^  +  Mc"-  +  Fx^  +  etc.,  (1) 

in  which  x  is  a  variable,  and  A,  B,  C,  etc.,  are  constants. 

Differentiating  (1),  we  have 

?^  =  Bdx  +  2Cxdx  +  ^Dx^dx  +  ^Ex^dx  +  ^Fx^dx  +  etc. 

\  +  X 

Dividing  by  dx, 

-^^  =B  +  2Cx  +  ZDx''  +  4^3  +  ^Fx'  +  etc.  (2) 

Differentiathig  (2),  and  dividing  by  dx,  we  have 

-  m  y—^—  =  2C7  +  2  •  ZDx  +  Z  A:Ex^  +  4.  bFx^  +  etc.  (3) 

(1  -\-xy 

Differentiating  (3),  and  dividing  by  2  and  by  dx,  we  have 

m — ^ —  =  37>  +  3  4i^.c  +  2  3  5F«2  +  etc.  (4) 

(1  +  xY 

Differentiating  (4),  and  dividing  by  3  and  dx,  we  have 

_  m ^ — -  --=:  4JS;  +  4  5iP!c  +  etc.  (5) 

(1  ^-xY  ^  ' 

Differentiating  (5),  and  dividing  by  4  and  dx,  we  have 

We  have  now  gone  far  enough  to  enable  us  to  determine  the  coefficients  A, 
B,  C,  D,  E,  and  F,  and  these  will  probably  reveal  the  law  of  the  series. 

As  all  the  above  equations  are  to  be  true  for  all  values  of  x,  and  as  the  coeffi- 
cients A,  B,  C,  etc.,  are  constant,  t.  e.,  have  the  same  values  for  one  value  of  x  as 
for  another,  if  we  can  determine  their  values  for  one  value  of  x,  these  will  be 
their  values  in  all  cases.     Now,  making  x  =  0,  we  have,  from  (1),  A  —  log  1=0; 

*  Whftt  the  relation  of  the  modiiln?  to  the  base  is,  we  are  not  now  concerned  to  know  ;  it 
will  be  determined  liereaftcr. 

t  The  numb<'r  is  1  4  a; ;  hence  tiie  differential  is  m  times  tlie  differential  of  1  +  a^  divided  by 
the  number  \-^x. 

X  Of  course  the  student  will  observe  what  forms  the  sncceedinir  t-  rms  in  this  and  the  other 
similar  cases  would  have.    Thus  here  we  should  have  5-F  +  6  •  6C?iC  4  3  5  •  T/Zr'  +  etc. 


174  ADVANCED  COURSE  IN  ALGEBRA. 

from  (2),  B  —  m\  from  (3),  C=  —  \m  ;   from  (4),  D  —  ^m  ;  from  (5),  E=  ~  \m; 
from  (6),  F=  im.    These  values  substituted  in  (1)  give 

^^  0*y^  Ct^  3?'' 

log  (l  +  x)=  m{x  — ^  +  -__  +  -_  etc.), 

the  law  of  which  is  evident.     This  is  the  Logarithmic  Senes,  and  should  be  fixed 
in  memory. 

ScH. — The  Napierian  system  of  logarithms  is  characterized  by  the  modu- 
lus being  1  (m  =  1).     Hence  the  Napierian  logarithmic  series  is 

....  x^      X*      x^      a-' 

log(l+ar)=«--  +-3-J+5-  etc. 

190,  Cor.  1. —  The  logarithms  of  the  same  number  in  differoit 
systems  are  to  each  other  as  the  moduli  of  those  sy steins. 

This  is  evident  from  the  general  logarithmic  series.  Thus  the  logarithm  of 
1  +  «  in  a  system  whose  modulus  is  m,  is  expressed 

log„(l  +x)«  =  m(;r-^  +  ^'-?j?  +^-etc.): 
Ss        o        4        a 

and  the  logarithm  of  the  same  number  in  a  system  whose  modulus  is  m'  is  ex- 
pressed 

log«.(l  +  X)*  =  m'{x  -^  +  ^-^4-^-  etc.). 

Now,  as  the  number  (1  +  x)  is,  by  hypothesis,  the  same  in  both  cases,  x  is  the 
same.     Hence,  dividmg  the  members  of  the  first  by  the  corresponding  members 

of  the  second,  we  have  logmd  +  x)  ^  _m  ^ 

logaiXl  +  «)      m' 

101,  Cor.  2.-^JIavin(/  the  logarithm  of  a  number  in  the  Napierian 
system,  ice  have  but  to  multiply  it  by  the  modulus  of  any  other  system 
to  obtain  the  logarithm  of  the  same  number  in  the  latter  system. 

Or,  the  logarithm  of  a  number  in  any  system  divided  by  the  loga- 
rithm of  the  same  number  in  the  Napierian  system,  gives  the  modidus 
of  the  former  system. 

102,  I^voh, —  To  adapt  the  Napierian  logarithmic  series  to  nu- 
merical computation  so  that  it  can  be  conveniently  used  for  comjyuting 
the  logarithms  of  numbers. 

/J.8  ^3  ^4  /j»5 

Sol. — That  log(l  -v  x)  =  x—  —  +  — —  t'^~F  —  ^^^->  ^^  ^°*  ^"  ^  practica- 

2         o         4         o 

ble  form  for  computing  the  logarithms  of  numbers  will  be  evident  if  we  make 

the  attempt.     Thus,  suppose  we  wish  to  compute  the  logarithm  of  3.     Making 


*  The  subscripts  m  and  m'  are  need  to  distinsrnish  between  the  system?,  as  log  (1  -t  O")  is  not 
the  sann  in  one  syptem  as  in  the  oihcr.  Ilead  log»»(l  +  a;),  "logarithm  of  \-^x  in  a  system 
whose  modulus  is  m.^''  etc. 


COMrUTATION   OF   LOGARITHMS. 


175 


2*      2'*      2*      2' 
a:  =  2,  we  have  ]og(l  +  2)  =  log  3  =  2  —  —  +  tt  —   r  +  ^    —  etc.,  a    series 

,6  o  4         5 

in  which  the  terms  are  growing  larger  and  larger  (a  diverging  series). 

We   wish   a   series   in   which  the   terms   will  grow  smaller  as   we  extend 

it  (a  converging  series).      Then  the   farther  we  extend  the   series,  the  more 

nearly  shall  we  approximate  the  logarithm  sought.     To  obtain  such  a  series, 

substitute  —x  iorx  in  the  Napierian  logarithmic  series,  and  we  have 

,      ,.        .  x'^      a?'      X*      x^ 

logil-x)=-x--- ------ etc. 

Subtracting  this  from  the  former  series,  we  have 

'1+x 


(1  +3'\ 
— —  j  =2{x  +  ^x^  +  W  +^x^  +  etc.). 


Now  put     x  =  - -,  whence   l+ic=l  +  ir 

^  224-1  2s+l 


2g  +  2 

22+1 


l-x  = 


2z 


22  +  1 


,  and 


posmg, 

l0g(l+2)  =  l0g2  +  2(^ 


= .     Hence,  as  log  ( j  =  log  (1  +  2)  —  log  2,  substituting,  and  trans- 

This  series  converges  quite  rapidly,  especially  for  large  values  of  2,  and  is 
convenient  for  use  in  computing  logarithms. 

103,  I^rob, —  To  compute  the  N'apierian  logarithms  of  the  natural 
numbers  1,2,  3,  4,  etc.y  ad  libitum. 

Solution. — In  the  first  place  wo  remark  that  it  is  necessary  to  compute  the 
logarithms  of  prime  numbers  only,  since  th3  logarithm  of  a  composite  number 
is  equal  to  the  sum  of  the  logarithms  of  its  factors  {17 S). 

Therefore  beginning  with  1,  we  know  that  log  1=0  (173). 

To  compute  the  logarithm  of  2,  make  2=1,  in  series  (A),  and  we  have  log  (1  + 1) 

-  log  1  =  log2  =  2(g  +  3^.+5^>  +  74''  +  9ir"  +  TT^-  +  i3^  +  i5V'+^''')- 

The  numerical  operations  are  conveniently  performed  as  follows : 
3      2.00000000 


.GG6G6G67 

1 

.07407407 

3 

.00823045 

5 

.00091449 

7 

.00010161 

9 

.00001129 

11 

.00000125 

13 

.00000014 

15 

.66660667* 

.02469136 

.00164609 

.00013064 

.00001129 

.00000103 

.00000009 

.00000001 

log  2  =  .69314718* 


*  Thongh  the  decimal  part  of  a  logarithm  U   generally  not  exact,  it  is  not  customary  ta 
annex  the  +  «»ign. 


176 


ADVANCED   COURSE  IN   ALGEBRA. 


Second.    To  find  lo^  3,  make  s  =  2,  whence 

log  3  =  log  2+2(1+ Jg-.+gl^.+Jj,+  J5j+  etc). 


Computation.       5 


25 
25 
25 
25 


2.00000000 


.40000000 

1 

.01600000 

3 

.00064000 

5 

.00002560 

7 

.00000102 

a 

.40000000 
.00533333 
.00012800 
.00000366 
.00000011 


.40546510 
log  2  =    .69314718 


/.  log  3  =  1.09861228 


Third.    To  find  log  4.    Leg  4  =  2  log  2  =  2  x  .69314718  =  1.38629436 
Fourih.    To  find  log  5.    Lei  z  =  4,  whence 

log  o  =  log  4  +  2(1  +  ^.  +  ^,  +  ^,+  etc.). 


Computation.       9 


81 
81 
81 


2.00000000 


.22222222 
.00274348 
.00003387 
.00000042 


.22222222 
.00091449 
.00000677 
.00000006 


.22314354 
log  4=  1.38629436 


.-.  log  5  =  1.60943790 

In  like  manner  we  may  proceed  to  compute  the  logarithms  of  the  prime  num. 
bers  from  the  formula,  and  obtain  those  of  the  composite  numbers  on  the  prin- 
ciple that  the  Ic^rithm  of  the  product  eqaals  the  sum  of  the  logarithms  of  the 
factors. 

Thus,  the  Napierian  Ic^rithm  of  the  base  of  the  common  system,  10,  =  log  5 
+  log  2  =  2.30258508. 


194,  Prop, —  The  modithts  of  the  common  system  is  .43429448  +  . 

Dem. — Since  the  logarithm  of  a  number,  in  any  system,  divided  by  the  Na- 
pierian logarithm  of  the  same  number  is  equal  to  the  modulus  of  that  system 
(191),  we  have 

— — '■ — ?- —  =  modulus  of  common  system. 
Nap.  log  10 


TABLES   OF  LOGARITHMS.  177 

But  com.  log  10  =  1,  aucl  Nap.  log  10  =  2.30258508,  as  found  above.     Hence, 
ModtUus  of  common  system  =  -^r^f^^^-^iTx  =  .43429448. 


Tables  of  Logarithms. 

19S,  As  one  of  the  most  important  uses  of  logarithms  is  to 
facilitate  the  performance  of  multiplication,  division,  involution,  and 
evolution,  when  the  numbers  are  large,  according  to  (178-lSl), 
it  is  necessary  to  have  at  hand  a  table  containing  the  logarithms 
of  numbers.  Such  a  table  of  common  logarithms  is  usually  found 
in  treatises  on  trigonometry  and  on  surveying,  or  in  a  separate 
volume  of  tables.*  These  tables  usually  contain  the  common  loga- 
rithms of  numbers  from  1  to  10000,  Avith  provision  for  ascertaining 
therefrom  the  logarithms  of  other  numbers  with  sufficient  accuracy 
for  practical  purposes.  Four  pages  of  such  a  table  will  be  found 
at  the  close  of  this  volume. 


196,  J^vob, — Tofiiid  the  logarithm  of  a  number  from  the  table. 

Solution. — The  logarithm  of  any  number  from  1  to  100  inclusive  can  be 
taken  directly  from  the  first  page  of  the  table.  Thus  log  2  =  0.301030,  and 
log  21  =  1.322219.t 

To  find  the  logarithm  of  any  number  from  100  to  999  inclusive,  look  for  the 
number  in  the  column  headed  N,  and  opposite  the  number  in  the  first  column  at 
the  right  is  the  mantissa  of  the  logarithm.  The  characteristic  is  known  by 
{185).    Thus  log  182  =  2.260071 ;  log  135  =  2.130334. 

To  find  the  logarithm  of  any  number  rej^resented  by  4  figures,  find  the  first  3 
left-hand  figures  in  column  N,  and  opposite  this  at  the  right  in  the  column  which 
has  the  fourth  figure  at  its  head,  will  be  found  the  last  four  figures  of  the  niau- 
The  other  two  figures  of  the  mantissa  will  be  found  in  the  0  column,  oppo- 


*  Mathematicians  and  practical  computers  generally  use  more  complete  and  extended  table:* 
than  those  found  in  connection  with  such  elementary  treatises.  The  common  tables  give  five 
places  of  decimals*  in  the  mantissa.  Those  in  connection  with  this  series  give  six.  Callet's 
tables  edited  by  Haslcr  are  standard  eight-place  logarithms.  Vega's  tiiblcs  are  among  the  best. 
Dr.  Bremiker's  edition,  translated  by  Prof.  Fischer,  is  a  favorite.  KOiilcr's  edition  of  Vega's 
contains  Gaussian  logarithms.  Vega's  tables  are  iseven-place.  Ten-place  logarithms  arc  neces- 
sary for  the  more  nccurnto  astronomical  calculations.  Prof.  J.  Mills  Peirce,  of  Harvard,  has  re- 
cently issued  nn  elegant  little  folio  edition  of  tables  containing  among  other  tilings  a  table  of 
three-place  logarithms  whicli  is  very  convenient  for  most  uses. 

t  This  page  is  really  unnecessary,  since  nothing  ran  be  found  from  it  which  cannot  be  found 
with  equal  case  from  \\\^  succeeding  part  of  the  table.  Thus,  the  mantissa  of  log  %i&  the  eamo 
as  the  mantissa  of  log  200 ;  and  the  mantissa  of  log  21  is  the  same  as  thai  of  lo^  210. 

12 


178  ADVANCED  COURSE  IN  ALGEBRA. 

Bite  the  first  three  figures  of  the  number  or  just  above,  unlesd  heavy  dots  have 
been  passed  or  reached  in  running  across  the  page  to  the  right,  in  which  case  the 
first  two  figures  of  the  mantissa  will  be  found  in  the  0  column  just  below  the 
number.  The  places  of  the  heavy  dots  must  be  supplied  witli  0'.-;.  The  charac- 
teristic is  determined  by  {183).  Thus  log  1316=3.119250 ;  log  2012=3.310056  ; 
log  1868  =  3.271377. 

To  find  the  logarithm  of  a  number  represented  by  more  than  4  fignres.  Let 
it  be  required  to  find  the  logarithm  of  1934261.  Finding  the  mantissa  correspond- 
ing to  the  first  four  figures  (1934)  as  before,  we  find  it  to  be  .286456.  Now  in  the 
pame  horizontal  line  and  in  the  column  marked  I),  we  find  225,  which  is  called 
the  Tabular  Difference.  This  is  the  difference  between  the  logarithms  of  two 
consecutive  numbers  at  this  }X)int  in  the  table.  Thus  225  (millionths)  is  the 
difference  between  the  logarithms  of  1934  and  1935,  or,  ns  we  are  using  it, 
between  the  logarithms  of  1934000  and  1935000,  which  differences  are  the  same. 
Now,  assuming  that,  if  an  increase  of  1000  in  the  number  makes  an  increase  of 
225  (millionths)  in  the  logarithm,  an  increase  of  261  in  the  number  will  make  an 
increase  of  -|2i|i,V,,  or,  .261,  »)f  225  (millionths)  in  the  logarithm,*  we  have  .261 
X  225  (millionths)  =  59  (millionths),  omitting  lower  orders,  as  the  amount  to  be 
added  to  the  logarithm  of  1934000  to  produce  the  logarithm  of  1934261.  Adding 
this  and  writing  the  characteristic  {185)  we  have  log  1934261  =  6.28G515.  In 
like  manner  the  logarithm  of  any  other  number  expressed  by  more  than  four 
figures  may  be  found. 

197,  Sen. — As  the  mantissa  of  a  mixed  integral  and  decimal  fractional 
number,  or  of  a  number  entirely  decimal  fractional,  is  the  same  as  that  of  an 
integral  number  expressed  by  the  same  figures  {184),  we  can  find  the  man- 
tissa of  the  logarithm  of  such  a  number  as  if  the  number  were  wholly  inte- 
gral, and  determine  the  characteristic  by  {185). 

198,  I^rob. — To  find  the  number  corresponding  to  a  given 
logarithm. 

Solution. — Let  it  be  required  to  find  the  number  corresponding  to  the  log- 
arithm 4.2vJ4567.  Ix)oking  in  the  table  for  the  next  kfss  mantissa,  we  find  .234517, 
the  number  corresponding  to  which  is  1716  (no  account  being  taken  as  to 
whether  it  is  integral,  fractional,  or  mixed  ;  as  in  any  case,  the  figures  will  be  the 
same).  Now,  from  the  tabular  difference,  in  column  D,  we  find  that  an  increase 
of  253  (millionths)  upon  this  logarithm,  would  make  an  increase  of  1  in  the 
number,  making  it  1717.  But  the  given  logarithm  is  only  50  greater  than  the 
logarithm  of  1716 ;  hence,  it  is  assumed  (though  only  approximately  correct) 
that  the  increase  of  the  number  is  -/s^  of  1,  or  .1976  f .  This  added  (the  figures 
annexed)  to  1716,  gives  17161976  -\- .  The  characteristic  of  the  given  logarithm 
being  4,  the  number  lies  between  the  4th  and  5th  powers  of  10,  and  hence  has  5 
integral  places,  .*.  4.234567  =  log  17161.976 +.  In  like  manner  the  number 
corresponding  to  any  logarithm  can  be  found. 


*  This  atsir.i  ption,   thotig'i   not  et  iitly  correct,   is  snfficluntly  accurate  for  all  ordinary 
purposes. 


COMPUTATION   BY  LOGARITHMS.  179 

199.  JProp.—  The  rnqyierlan  base  is  2.718281828. 

Dem. — Let  e  represent  the  base  of  the  Napierian  system.     Then  by  {190) 
com.  log  e  :  Nap.  log  e  :  :  .43439448  :  1. 
But  the  logarithm  of  the  base  of  a  system,  taken  in  that  system  is  1,  since 
a'  =  a.     Hence,  Nap.  log  e  =  l,  and  com.  log  e  =  .43429448.     Now  finding  from 
a  table  of    common  logarithms  the  number  corresponding  to  the  logarithm 
.43429448,  we  have  e  =  2.718281828. 


Examples. 

1.  If  3  were  the  base  of  a  system  of  logarithms,  what  would  be  the 
logarithm  of  81  ?  Of  729  ?  If  5  were  the  base,  of  what  number  would 
3  be  the  logarithm  ?     Of  what  2  ?     Of  what  4  ? 

2.  If  2  were  the  base,  what  would  be  the  logarithm  of  J  ?     Of  ^  ? 

3.  If  16  were  the  base,  of  what  number  would  .5  be  tlic  logarithm  ? 
Of  what  .25  ? 

4.  Ill  the  common  system  we  find  that  log  156=2.193125.     Show 

I.JLJL3126 

that  this  siguifies  that  10^«"«^"  =156. 

5.  Log  1955=3.291147.  To  what  power  does  this  indicate  that 
10  is  to  be  raised,  and  what  root  extracted  to  make  1955  ? 

6.  Find  from  the  table  at  the  close  of  the  volume  what  root  of 
what  power  of  10  equals  2598. 

7.  Multiply  1482  by  136  by  means  of  logarithms,  using  the  table 
at  the  close  of  the  volume.     (See  178,) 

8.  Perform  the  following  operations  by  means  of  logarithms: 
116.8  X  1879;  2769 -r  187;  15.13  x  1.3476;  257.16  -^  18.5134; 
.126  H- 6.1413;  .11257  x  .00126;  (1278.6)'^;  (112.37)'. 

9.  Perform  the  following  operations  by  means  of  logarithms:  ^2 
to  5  places  of  decimals ;  y  5  to  3  places  of  decimals ;  y  2341564273 
to  two  places  of  decimals ;  a/3015618  to  4  places  of  decimals. 

10.  Perform  the  following  operations  by  means  of  logarithms: 
V^.01234  to  4  places  of  decimals;  a/.03125  to  5  places  of  decimals* 
V^0002137  to  5  places  of  decimals. 

SuG's.— Log  .01234=2.091315.  Now  to  divide  this  by  3.  we  have  to  remember 
that  the  characteristic  alone  is  negative,  i.  e.  that  2.091315  =- 2  I-. 091315,  ox 


180  ADVA^'CED    COUKSE  IN   ALGEBUA. 

—  1.908685,  whidi  is  all  negative.  Dividing  this  by  3,  we  have  —.636228,  or 
0— .636228=1.363772.  But  a  more  convenient  way  to  effect  the  division  is  to 
write  2.091315  =  3  + 1.091315,  and  dividing  the  latter  by  three  we  obtain 
1.363772,  in  which  the  characteristic  alone  is  negative,  thus  conforming  to  the 
tables. 

To  divide  13.341652  by  4,  we  write  for  13.341652,  -16+3.341652,  and  dividing 
the  latter  obtain  4.835413. 

11.  Divide  as  above  11.348256  by  3;    17.135421  by  5;    1.341263 
by  6. 

12.  Given  the  following  to  compute  x  by  logarithms : 
201.56 :  134.201 : :  18.654 :  x;  2350.64  :  .212  : :  1.1123  :  x ; 
X :  234.008  : :  15.738  :  200.56 ;            123  :  a: : :  2.01 :  .03. 


/fit /pt 

13.  Having  i/  =  A/  — — —  to  express  the  equivalent  operations 


in  losrarithms. 


—  rt)  («  —  h)  (s  —  c) 


SuQ'8.     y  =  V{a  -  X)  {a  +  .t)-i-(1  +  x).     .-.  log  y=  \  [log  {a  -  x)  +  log  {a+x) 
-\og{l+x)l 

2  1 

14.  Given  y=x^ (l—.v^)^  to  express  the  equivalent  operations  in 
logarithms.    Also  //  =  A/  j-.    Also  //  =  A/ ■ 

Also   1/  = rir— •      Also  ^  =  i  /  — -  .      Also  giveu    Ti  ••  -7  :  : 

\/m^—X' :  y  to  express  log  y. 

15.  Differentiate  y  =  \og{a^  —x*), 

Sug's. — Write  y  =±  log  (a  +  a^)  +  log  (a  —  x).     Then  differentiating,  we  have 

mdx       mdx    _    ,._        ^.  ^.         .^i      x^  _^    •  t        j       d{a*~x^)* 

dy  = .  Or  differentiating  without  factoring,  we  have  ay  =—^-5 5- 

(ti  -k'  X     a  —  X  a  — X 

iimxdx 

=  —  --i 5.    WHien  reduced  the  results  are  the  same,  but  the  former  is  usualh' 

«'— a;* 

the  more  elegant  method. 

16.  Differentiate   the   following:     y  =  log  (1  —  a:) ;     y  —  log  ax; 

y  =z  log  x^  ;  y  =  \o^     ;  //  =  log  v^l  +  .r. 

♦  This  form  signifies  that  a'  -x^  is  to  be  differentiated.     The  operation  is  only  indicated,  r.ot 
performed. 


SUCCESSIVE  DIFFEKENTIATION.  181 

Sug' 8.— Remember  that  log  x"^  =  3  log  a; ;  and  also  that  log  /y/l  -\- x  — 
Hog(l+a). 

17.  Find  from  the  table  at  the  close  of  the  volume  that  Kap.  log 
1564=7.3550018.  Find  in  like  manner  the  Napierian  logarithms  of 
5,  120,  and  2154372. 

18.  Knowing  that  the  Napierian  logarithm  of  22  is  3.0910425,  how 
would  you  find  the  common  logarithm  of  23  from  the  logarithmic 
series  {192)  ? 

19.  The  common  logarithm  of  25  is  1.39794.  What  is  the  modu- 
lus, and  what  the  base  of  a  system  which  makes  the  logarithm  of 
25    2.14285? 

Query. — How  do  you  see  at  a  glance  that  the  required  base  is  a  little  less 
than  5  ? 


SECTION   V. 

SUCCESSIVE    DIFFERENTIATION,   AND   DIFFERENTIAL 
COEFFICIENTS. 

200,  JProp, — Differentials^  though  itifinitesimah,  are  not  neces- 
sarily equal  to  each  other. 

Dem. — ThuB,  let  11— ^x"^.  Then  dt/=6x^dx.  Now,  for  all  finite  values  of  x, 
dy  is  an  infinitceinial,  Binco  no  finite  number  of  limes  ibe  infinitcf^imal  dx 
can  make  a  finite  quantity,  and  dy  is  6x'*  times  dx.  But  for  ,^'=l,  dy  is  G  times 
dx  ;  for  x=2,  dy  is  24  times  dx;  for  a;=8,  dy  is  54  times  dx. 

201*  Coil. —  When  y  =  f(x),  dy  18  got er ally  a  variable,  and  hence 
can  he  differentiated  as  any  other  variable. 

202.  NoTATiO]!?. — The  differential  of  dy  is  written  d^y,  and  read 
"  second  differential  of  y."  The  differential  of  d^y  is  written  d'^y,  and 
read  "third  differential  of  y,"  etc.  The  superiors  2  and  3  in  such 
cases  are  not  of  the  nature  of  exponents,  as  the  t?  is  not  a  symbol  of 
number. 

203,  In  differentiating  y=:f(x)  Bticcessively,  it  is  customary  to 
regard  dx  as  constant.  Tliis  is  conceiving  x  to  change  (grow)  by 
equal  infinitesimal  increments,  and  thence  ascertaining  liow  y  raries. 
In  general,  y  will  not  vary  by  equal  inciements  when  x  does,  as 
appears  from  the  demonstration  above. 


182  ADVANCED  COURSE  IN  ALGEBRA. 

204,  A  Second  differential  is  the  difference  between  two 
consecutive  states  of  n  first  differential. — A  Third  Differential 

is  the  difference  between  two  consecutive  states  of  a  second  differ- 
ential, etc. 

III.— In  the  function  y-2.c',if  x  passes  to  the  next  state,  we  have  dy—Qx^dx. 
Now  dy,  though  an  infinitesimal,  is  still  a  variable,  for  it  is  equal  to  Qdx  times 
a-*,  and  ^  is  a  variable.  Hence  if  ,r  takes  an  infinitesimal  increment,  di/  will  pass 
to  a  consecutive  state.  In  other  words,  we  can  differentiate  di/=Gdx.v^,  just  as 
we  could  It  =  j?ix^,  dy  being  a  variable  function.  M.v  a  constant  factor,  and  x  the 
variable.  Representing  the  differential  of  dy  by  d-y,  wo  have  d'^y  =  6d?;  2.idx, 
or  d^y=l'ixdx',  dx-  being  the  square  of  dx,  not  the  differential  of  x\  To  indi- 
cate the  latter  we  would  write  rf(.r*). 


JEXAMPLES. 

1.  Given  y  =  Sj^  —  2a^  to  find  the  third  differential  of  i/,  or  (Py. 

Solution. — Differentiating  y=3.r'— 2.r*,  we  have  rfy=15.c*rfx— 4r(f.c.  Now, 
regarding  dx  as  constant,  and  differentiating  again,  we  have  d^y=60x*dx' 
—Adx^*  Differentiating  again  in  like  manner,  we  obtain  d*y=180x^dx^,  the 
second  term  disappearing,  since  4dx*  is  constant. 

2.  Given  y  =  2a^  —  Sx  -{-  5  to  find  the  second  differential  of  ?/, 
t.  c.  d^y. 

3.  Given  y  =  [x  —  af  to  find  the  third  differential  of  ^. 
SuQ'8.    dy=3(x-aydx.  d*y=6(x-a)dx',  d^y=Ux^. 

4.  Given  y  =  Ax  +  Bx"  +  Cct"  +  Dx\  to  find  the  4th  differential 
of  y,  J,  B,  C,  and  D,  being  constiint.  (ty  =  4  •  3  •  2  Ddx*. 

5.  Differentiate  y  =:  A  +  Bx  +  Cj^  -{-  Dx"  +  Ea^  -\-  Fa^  -\-  etc.,  5 
times  in  succession. 

G.  Differentiate  y  =  {x  —  \){x  —  2) (a:  —  3) (a:  —  4)  twice  in  suc- 
cession without  expanding. 

SoG'8.    dy  =  {x-%){x-Z){x-^)dx  +  {x-\){x-S){x-^)dx+{x-l){x-2){x-A) 
dx-{-{x-\){x-2){x-'S)dx. 

=  [(.r-2)  {x-2)  (3--4)  +  {x-\)  ix—Z)  {x-i)  +  ix-\)  {x-2){x-'i) 
^{x-\){x-2){x-2,)]djc. 

d'^y  =  \{x-Z){x-\)dx+{x-2){x-^)dc^{x-'^){x-Z)dx  +  {x-Z){x-^)dx  +  {x-\) 
(x-^)dx  +  {x-\)  (a;-3)  dx-^{x-2)  (.r-4)  dx+{x-\)  (.r-4)  dx+{x-\){x-2) 
dx  +{x-^)  {x-'S) (ic+(.c-l)  (.r-3)  dxH-t-X)  (.c-2)dc]dr. 

*  To  differentiate  \bx*dx.  calling  dx  constant,  we  may  write  15cto  x*.  Now  15dx  is  con- 
stant. Hence  differentiating  a;4,we  have  Ax^dx,  which  multiplied  by  the  constant  15tfx,  gives,  ai 
above,  BOx'ctr^.    The  dx^  is  "  the  sqnarc  of  rfr,"  not  t'ac  cliflfereniial  of  x"^. 


DIFFERENTIAL  COEFFICIENTS.  183 

=  [(^-3)  (^-4)  +  {.v-2)  (,i--4)  +  (.c-2)  (x-'S)  +  (x-d)  (x-i)  +  (x-l)  {x-4) 
+  (^-1)  (^-3)  +  {x-2)  (x-i)  +  (x-l) (x-i)  +  (x-l)  {x-2)  +  {x-2)  {x-3) 
+  (x-l)  {x-'S)  +  {x-l){x-2)]dx'. 

7.  As  above,  differentiate  y  =  (x  —  a){x  —  b)(x  —  c)  twice  in  suc- 
cession without  expanding. 


Differential  Coefficients. 

205.  The  First  Differential  Coefficient  is  the  ratio  of 
the  differential  of  a  function  to  the  differential  of  its  variable.     Thus, 

if  y=f(x),   and   (lij=f'{x)dx,  ^-^  =  f'(x),  and  ^,  or  its  equivalent 

f'{x),  is  the  first  differential  coefficient  of  ;y,  oy  f(x). 

III. — The  meaning  of  this  is  simple.     Thus,  if  y  =  2x^,  -~-  =  8x^  ;  that  is,  if 

X  takes  an  infinitesimal  increment  dx,  y  takes  an  infinitesimal  increment  dy, 
which  is  to  dx,  as  8.i' '  is  to  1,  or  the  ratio  of  dy  to  dx  is  8.f '.  In  still  other  words, 
y  increases  8.c '  times  as  fast  as  x.  The  reason  for  calling  this  a  differential 
coefficient,  is  that  it  is  the  coefficient  by  which  the  increment  {dx)  of  the  variable 
must  be  multiplied  to  give  the  increment  {dy)  of  the  function. 

206,  The  Second  Differential  Coefficient  is  the  ratio  of 
the  second  differential  of  a  function  to  the  square  of  the  differential 
of  the  variable.     Thus,  if  y—f{x)y  dy=f{x)dx,  and  d'^i/=f"{x)dx'^, 

— ^=/"(rc),  -y^  or  its  equivalent /"(a:),  is  the  second  differential  coef- 
ficient of //,  or  f(x).  In  like  manner  Third,  Fourth,  etc.,  differential 
coefficients  are  the  ratios  respectively  of  the  third,  fourth,  etc.,  dif- 
ferentials of  a  function,  to  the  cube,  fourth  power,  etc ,  of  the  dif- 
ferential of  the  variable.  Thus,  if  y=f(x),  dy=f\x)dx,  dhj—f'ix  rZr  2, 
d^y=^f"'(x)dx^,  and  d'^y  =/'" {x)dx^,  the  successive  difCercntitil  coclli- 

cientsare  f=}  {x),    J=/  (x),  ^=f    ix),  and  ^,=/'\.r). 

III. — Too  much  pains  cannot  be  taken  by  the  student  in  order  to  get  a  clear 
conception  of  the  meaning  of  the  various  symbols  f{x),  f'{-i'),  /"(•')»  f"'i-^)'  ^^^- 

To  illustrate,  suppose   we  have  y  =  2.c^— a;'H-C,  whence -p  =  8.i;'—C.t;'-,  - -— 

*  To  produce  the  pnccessive  differential  coefBcients  wc  maj'  produce  the  corrcspnndin;?  guc- 
cessive  differentials  as  in  the  preceding  example?,  or  wc  may  proc.;c  1  thn^*:  ■_='\r3-3x2  can 
be  differentiated,  remembering  that  dy  is  variable  and  dx  constant,  and  it  gives  ^— =24x2(/a? 

(fiu 

-%xdx,  whence  ;i-^'=24a:'-6a:. 


184  ADVANCED  COUKSE  IN  ALGEBRA. 

d  ii  d^  7/ 

=  24c*— dr,  -r-^  =  48a'— 6,  and  -j—  =48.     Now  in  this  case  y  =f(x),  i.  e.,  v  is  a 

dv 

function  of  .r  ;  so  -r^  is  also  a  function  of  x,  being  equal  to  8^''- 3a''  ;  but,  as  it 

is  not  the  same  function  of  r  that  y  is,  we  call  it  the  /  prime  function,  and  write 

diJ  ft  i/  ft^ fj 

~  =  f'U).    In  like  manner  -~^  =f"(x)  means  that  -~  is  some  function  of  ar, 

dx     ^  ^  '  dx*      ''    ^  '  dx* 

but  a  different  one  from  either  y,oT~  .    It  may  be  observed  that,  in  t?iis  example, 

d*y 

3-j  is  not  a  function  of  x,  and  hence  the  inquiry  arises  as  to  the  propriety  of  the 

d*u 
notation  ^-^  =/'*  (x).    It  must  be  remembered  that  this  form  of  notation  is  the 

d*v 
general  form,  and  it  is  the  general  fact  that  -j-^  is  a  function  cf  .r,  though  in 

special  cases  it  may  not  be. 


Examples. 

1.  Produce  the  1st,  2cl,  3(1,  and  4th  differential  coeflRcients  of 
y=a:»— 3a:»  +  .T— 10. 

Operation,    dy  =  5x*dx  —  Qx'dx  +  dx,  whence  ■—  =  5x*  —  9a;*  +  1.    Differ- 

dx 

entiating  the  latter*  -r^  =  2(te^da5  -  18a-da?,  whence  -^  =  20x^  -  ISa-.     Again 
ax  (tx 

differentiating,  ^  =  (60a;*  -  18)dc,  whence  ^  =  OOc*  -  18.      Finally,   ^ 
ax  ax'  ax 

=  12ar. 

2.  If  y =00:*— 3a;,  what  is  the  ratio  of  the  increase  of  y  to  that  of  a;, 
in  general  ?     What  is  it  when  x=-\  ?     When  a:=2  ?    AVhen  a:=3  ? 

Ans,  In  general,  y  increases  10a:— 3  times  as  fast  as  x.  When 
a:=l,  y  is  increasing  7  times  as  fast  as  x.  When  a:=2,  y  is  increas- 
ing 17  times  as  fast  as  x, 

3.  If  7/=x*4-2a:*— a:  +  10,  what  is  the  ratio  of  the  3d  differential  of 
?/ to  the  cube  of  the  differential  of  .^•  ?  WHiat  is  it  when  .^=1? 
When  x=^  ?    ATlien  a:=|  ?    What  is  the  name  of  this  ratio ? 

4.  If  ?/=(rt  +  a:)'%  what  is  the  1st  differential  coefficient  of  tlie  func- 
tion? What  the  2d?  What  the  3d?  AVhiit  the  5th?  What 
the  11th? 

^  =  j»(M-l)(?;i-2)(m-3)(m-4)(^<-f-a:)'"-». 
*  See  foot-uote  on  preccdliis  paje. 


Taylor's  formula.  185 

5.  Produce  the  first  5  successive  differential  coefficients  of 

207,  Sen. — The  successive  differential  coefficients  of  a  function  of  the 
form  A+Bx+Cx--\-Dx^-\-  etc.,  or  .^«^-^a;«-^ +^a;"-2+  etc.,  are  readily  writ- 
ten by  inspection.  Thus,  CfxW  x* —2x^+bx^ +x—\2,f{x).  Let /'(.!•)  mean  the 
first  differential  coefficient,  f"{x)  the  second,  f"'{x)  the  third,  etc.     We  have 

f{x)  =  x^  -  2^='  +  5.r^  +  X  -  13. 
/'(.r)  =  4c'  -  6a;2  +  lOi;  +  1. 
f"{x)  =  12.C2  _  12^  ^  10. 
/'"(.y)  =  242;  -  12. 
/'^(i)  =  24. 
/v(.r)  =  0.    Here  the  processes  terminate. 

Each  of  the  above  is  produced  from  the  preceding  by  multiplying  the 
coefficient  of  x  in  each  term  by  the  exponent  of  x  in  that  term  and  diminish- 
ing the  exponent  by  1. 

6.  According  to  the  method  indicated  in  the  last  scholium,  write 
out  the  successive  differential  coefficients  of  the  function  2x^-\-Sx* 
-5a:'  +  10.    Also  of  2^-32:'"'  +  ^''.     Also  of  3  +  2:c-4a;H3^. 


SECT/ON  VL 
TAYLOR'S    FORMULA. 


208.  Def. — Taylor^s  Formula  is  a  formula  for  developing 
a  function  of  the  sum  of  two  variables  in  terms  of  the  ascending 
powers  of  one  of  the  variables,  and  finite  coefficients  which  depend 
upon  the  otlier  variable,  the  form  of  the  function,  and  its  constants. 

209,  Def. — If  n  —fix  +  y),  i-  e.,  if  u  is  a  function  of  the  sum 

of  the  two  variables  x  and  y,  and  we  differentiate  as  though  one  of 

the  variables,  as  x  or  y,  was  constant,  the  differential  coefficients  thus 

formed  are  coW^di.  Partial  Differential  Coefficients,    The 

partial  differential  coefficients  of  u,  when  x  is  considered  variable 

^   ^   ^,  du     cl^u     cfu     (fn 

and  y  constant,  are  represented  thus:       t-,    ^,    -r^,   -7^4,  etc. 

When  y  is  considered  variable  and  x  constant,  we  write  the  coeffi- 

.     ,       du     d\i      d^u      dSi      , 
cients      v",     3—2,     -7-1,     -J-.,  etc. 
dy      dy''     df      dy* 


186  ADVANCED   COUllSE  IN  ALGEBRA. 

210.  Lemma. — If  u  =  f(x  +  y),  the  partial  differential  coeffi- 
cients   T-    and   T-    are  equal. 
dx  dy 

Dem. — Having  u  =:J{x  +  y),  if  x  take  an  increment,  we  have  u  +  dxU* 
=J[z  +  dx  +  y)  =  /[(.c  +  y)  +  dx] ;  whence  d^u  =  f  [{x  +  y)  +  dx]  -  f{x  +  y), 
eince  a  differential  is  the  difference  between  two  consecutive  states  of  the  func- 
tion. Again,  if  y  take  an  increment,  we  have  u  +  dyU  =  J\x  +  y  +  dy) 
=/[(«  +  y)  +  ^y] ;  whence  d„w  =/[(-»  +  y)  +  dy]  -  f{x  +  y).  Now  the  foi'm  of 
the  values  of  drU  and  dyU,  as  regards  the  way  in  wliich  x  and  y  are  involved,  is 
the  same ;  hence,  if  it  were  not  for  dx  and  dy,  they  would  be  absolutely  equal. 
Passing  to  the  differential  coefficients  by  dividing  the  first  by  dx  and  the  second 

by  dy.  we  have  "-^  =/[<^-*-y'+'^]-«^+y) .    and  'Jl!  =M^+y)+dl/]-Al±^) , 

''    "  dx  dx  dy  dy 

But,  in  differentiating,  the  differential  of  the  variable  enters  into  every  term ; 
hence /[(j;  +  y)  +  dx]  —f[x  +  y),  as  it  would  appear  in  application,  would  have 
&  dxin  each  term  which  would  be  cancelled  by  the  dx  in  the  denominator  in  the 

coeflBcient,  and  —  would  be  independent  of  dx.  In  like  manner  —  is  independ- 
ent of  dy.  Hence,  finally,  as  these  values  of  the  partial  differential  coefficients 
are  simply  functions  of  ( c  +  y),  of  the  same  form,  and  not  involving  dx  or  dy, 
they  are  equal,    q.  e.  d. 

III. — To  make  this  clear,  let    u  =  (x  +  yy.     Then  dxU  =  3(.c  +  yfdx,  or 

—  =S{x  +  y)*.  Again,  dyU  —8(x  -\-  y)*dy,  or  ^  —  3(.r+  y)".  Hence  we  see  that 
dx  dy 

_  ...  ,      ,  .    du         1  .  du         1  - 

So,  agam,  \t  u  =  log  (x  +  y),  -r  —  .  ^^^  t-  = 5     hence 

'    ^  ^^        '^^'  dx      x+y  dy      x  +  y' 


du 
dx'~ 

du 
~dy- 

du 
dx 

du 
~  dy' 

211.  I*rob. —  To  produce  Taylor^ s  Formida, 

SOLIJTION. — Let  u  =f{x  +  p)  be  iHie  function  to  be  developed.  It  ie  proposed 
to  discover  the  law  of  the  development  when  the  function  can  be  developed  in 
the  form 

u  =J{x  +  y)  =  A  +  By  +  Cy''  +  Dy'  +  Ey*  +  etc.,  (1) 

in  which  A,B,  C,  etc.,  are  independent  of  y,  and  dependent  on  x,  the  form  of  the 
function,  and  its  constants. 

Supposing  X  constant  and  differentiating  with  reference  to  y  as  variable,  re- 
membering that,  as  A,  B,  (7,-etc.,  are  functions  of  x,  and  not  of  y,  they  will  be 
considered  constant,  we  liave 

^  =B-^2Cy  +  SDy'  +  4^»  +  etc.  (2) 
ay 

*  As  we  are  to  consider  the  effect  produced  npon  u  by  an  increment  in  x.  and  al?o  by  an  in- 
crement in  y,  we  adopt  a  form  of  notation  to  distinguish  between  the  increments  of  //.  Thus 
rfj-»/ means  the  increment  which  w  takes  in  consequence  of  a;  having  talvcn  the  increment  dx, 
whMc  y  remained  constant.  So  cfy?<  represents  the  increment  of  m  consequent  upon  the  incre- 
ment dy  of  y. 


Taylor's  formula.  187 

Again,  differentiating  (1)  with  respect  to  x,  y  being  supposed  constant,  and  re- 
membering that  A,  B,  C,  etc.,  are  functions  of  x,  wc  have 

dx      dx         dx  dx  dx  dx 

Hence  by  {210) 
B+2Cy  +  dDy'  +  ^W\  etc.  =  ^  +  ^^  +  ^2/'  +  ^2/^  +  ~P'+  etc.   (4) 

Now,  by  the  theory  of  indeterminate  coefficients,  the  coefficients  of  tiie  like 
powers  of  y  are  equal,  and  we  have 

B  =  ^,    2(7=^,    3i)=^.    4:E=^,    etc. 
dx  dx  dx  dx 

But  as  (1)  is  true  for  all  values  of  y,  we  may  make  y  =  0;  whence 
A  =f(x)  =  u' ;  letting  u'  represent  the  value  of  the  function  u,  when  y  =  0. 
Now,  as  A  is  independent  of  y,  it  will  have  the  same  value  for  one  value  of  y  as 
for  another ;  hence  A  =f{x)  =  u'  is  the  general  value  of  A. 

dA  du' 

Aeain,  B  —  — .     But  as  J.  =  w',  a  function  of  x,  dA  —  du' ,  and  B  =  —-. 
^  dx  dx 

In  like  manner    2C  = — .      But  as    B  = —,    dB  =  dl-—)  = --—,    and 
dx  dx  \dx/        dx 

So,  also,as3D=  -.  and^(7  =  ^^(-,)  =  -^^,,i)  =  j^^. 

1  d^u' 
Similarly  we  find  E  =   —        ^  >  and  the  law  of  the  series  is  apparent. 

Finally,  substituting  the  values  of  A,  B,  C,  etc.,  in  (1),  we  obtain 

,       du   y       d^u'y"^       d'^u' y'^       d*u'    y*        ^         ,^, 
„=/(.  +  y)=«+^f+^|  +  ^|+^^+etc.,     (5) 

which  is  Taylor's  Formula. 

212,  SCH. — Taylor's  Formula  develops  u  —  f  (x  +  y)  into  a  series  in 
which  the  jirst  term  is  the  value  of  the  function  when  y  =  0  ;  the  second 
term  is  the  first  differential  coefficient  of  the  function  when  ^  =  0,  into  y ; 
the   third  term   is   the   second    differential    coefficient  of  the   function  when 

y  =  0,  into  ^  ;  etc.,  etc. 

If. 

As  u'  is  f(x+y)  when  y  =  0,  we  may  write /(a;)  for  u',  and  for  — ,  f'{x)  ;  for 
^^»  f"{x)  ;   for  ^»  /'"(«);  etc.,  as  before  explained.     The  formula  then  be- 

ClX  (IX 


*  Tne>«e  forms  arc  indicated  operations.    Thus,  as  ^  is  a  function  of  x.  when  we  difTerentiatc 
With  re?pcct  to  x  wc  write  dA,  and  to  pass  to  the  differential  coefficient  have  to  divide  by  dx. 


188  ADVANCED  COURSE  IN  ALGEBRA. 

u=f{x.^y)  =/(.r)+/'(.r)  |  +  f\x)  '^  ^f"\x) ^'  +/- {x) 'j^'  +  etc.  (6) 

This  is  a  very  important  method  of  writing  Taylor'a  Formula,  and  should  be 
clearly  understood,  and  firmly  fixed  in  memory. 


Examples. 

1.  Develop  (x-\-yY  by  Taylor's  Formula. 

Solution. — Putting    u  =  {x+yf,  we  have  u'  =  .t',  —t-  =  5x*,    -^  =  20a;', 

— ^  =  60^    — —  =  120a;,  and  — -  =  120.     Here  the  coefficients  terminate,  as 
dx>  '    dx*  dj* 

the  differential  of  a  constant  is  0. 

Substituting  these  values  in  (5)  {211),  or  {%).{212),  we  have 

u  =  (a;+y)»=  a;»+  5x*y  +  10a;='y«+  10a;*y'+  5ay*-}-y«. 

The  same  as  by  the  Binomial  Formula. 

2.  Develop  (x—yY  by  Taylor's  Formula,  and  compare  the  result 
with  that  obtained  by  means  of  the  Binomial  Formula.  Also  {x  +  y)  ^. 
Also  {x—y)~\    Also  (x-^y)  '. 

3.  Show  that 

n  =  log(.  +  y)=loga:+|-£  +  g-£  +  etc. 

4.  Develop  (x+y)"  by  Taylor's  Formula,  thus  deducing  the  Bi- 
nomial Formula. 


213»  Taylor's  Formula  is  much  used  for  developing  a  function 
of  a  single  variable  after  the  variable  has  taken  an  increment.  When 
so  used  the  increment  may  be  conceived  as  finite  or  infinitesimal, 
only  so  that  it  be  regarded  as  a  variable. 

Ex.  1.  Given  y  =  2x^  —  x* -\-  6x  —  11,  to  find  y',  which  represents 
the  value  of  the  function  after  x  has  taken  the  increment  Ji. 

Solution. — In  the  function  as  given,  we  have  p  =  fix),  and  are  to  develop 
^'=zf{x  +  //).    By  Taylor's  Formula  we  have 

dv  d^v 

From  y  =  2x^—X'  +  5a;  —  11,  we  have  -f-  —  (ix-  —  2x  +  5,    ~  =  12a;  —  2, 

dx  ux 


INDETERMINATE   EQUATIONS.  189 

— ^  =  12,  aud  subsequent  differential  coefficients  0.     Substituting  these  valuei 
in  the  formula,  we  obtain 

2  2    3 

=  2^''-.i'2+5«-ll  +(6x2-2^+5)7i+(6a;  -l)h^+2h\ 

This  result  is  easily  verified  by  substituting  x+h  for  x  in  the  value  of  y,  as 
given  in  the  example.     Thus, 

y'=%i:+hf-{x+hf-^b{x+h)-n; 
a  result  which  will  reduce  to  the  same  form  as  the  other. 

2.  Given  y=^x^—2x^,  to  develop  y\  the  value  of  y  when  x  takes 
the  increment  h. 


SECTION  VII, 
INDETERMINATE  EQUATIONS. 

214.  An  Indeterminate  Equation  between  two  quan- 
tities, as  X  and  y,  is  an  equation  which  expresses  the  only  relation 
which  is  required  to  exist  between  the  two  quun titles. 

III. — Suppose  we  have  2a;+8y=7,  and  that  this  is  the  only  relation  which  is 
7-equired  to  exist  between  x  and  y.     Then  is  %x-\-Zy=l  an  indeterminate  cqua- 

X 

Cion.     So  also,  if 6=:cy  is  the  only  relation  required  to  exist  between  x  and  y, 

0/ 

this  is  an  indeterminate  equation.  In  like  manner  y-  =  2x-^  —  'dx  is  an  in- 
determinate equation  if  it  expresses  the  only  relation  which  is  required  to  exist 
between  x  and  y. 

The  propriety  of  the  term  indeterminate  is  seen  if  we  observe  that  such  an 
equation  does  not  fix  the  values  of  x  and  y,  but  only  their  relation.  Thus,  in  the 
equation  2.7;  +  %  =  7,  ar  may  be  2,  and  y  1,  and  the  equation  be  satisfied.  So  x 
may  be  3,  and  y  ^,  and  the  equation  be  satisfied.  In  fact,  any  value  may  be 
assigned  to  one  of  the  quantities  and  a  corresponding  value  found  for  the  other. 
Hence  the  equation  does  not  determine  the  values  of  the  quantities. 

2 15.  An  equation  between  three  quantities  is  indeterminate  if  it 
expresses  the  only  required  relation  between  the  quantities,  or  if 
there  is  but  one  otlier  relation  required  to  exist. 

III. — Thus,  if  2x  +  Sy—5z—10  is  the  only  relation  which  is  required  to  exist 
between  x,  y,  and  z,  it  is  evident  that  the  equation  does  not  determine  particular, 
definite  values  for  x,  y,  and  z.  So  also  if,  in  addition  to  the  relation  expressed 
by  this  equation,  it  is  required  that  2x  shall  equal  fSy,  or  2a;— 6y,  these  two 


190  ADVANCED  COURSE  IN  ALGEBRA. 

equations  will  not  fix  tlie  values  of  .i;  y,  and  z.  For  if  2.r=6y,  the  former 
equation  becomes  9^—52=10,  which  may  be  satisfied  for  any  value  of  z,  and 
a  corresponding  value  of  y,  as  shown  above. 

216*  In  general,  if  there  are  n  quantities  involved  in  any 
number  of  equations  less  than  n,  and  these  are  the  only  relations 
required  to  exist  between  the  n  quantities,  the  equations  are  in- 
determinate. 

217>  In  indeterminate  equations  the  quantities  between  which 
the  relation  or  relations  are  expressed  are  properly  variables,  i.  e., 
they  arc  capable  of  having  any  and  all  values.* 

III. — Thus  in  the  indeterminate  equation  5y  —  Sx=  12,  any  value  may  be 
assigned  to  x,  and  a  corresponding  value  found  for  y;  or  any  value  may  be 
assigned  to  y,  and  a  corresponding  value  found  for  x. 

218,  There  are,  however,  many  classes  of  problems  which  give 
rise  to  equations  which  are  called  indeterminate,  although  they  are 
not  absolutely  so:  in  such  problems  there  is  some  other  condition 
imposed  than  the  one  expressed  by  the  equation,  but  which  con- 
dition is  not  of  such  a  character  as  to  give  rise  to  an  independent, 
simultaneous  equation.  Such  an  equation  may  have  a  number  of 
values  for  the  variables,  or  unknown  quantities,  involved,  but  not  an 
unlimited  number. 

III. — Let  it  be  required  to  find  the  podtite,  integral  values  of  x  and  y  which 
will  satisfy  the  equation  2x  +3y  =  35.  Now,  if  2a;  +  3y  =  35  were  the  only  rela- 
tion required  to  exist  between  x  and  y,  there  would  be  an  infinite  number  of 
values  of  each  which  would  satisfy  the  equation,  as  shown  above.  But  there  is 
the  added  condition  that  x  and  y  shall  be  positive  integers.  Tliis  greatly  re- 
stricts the  number  of  values,  but  does  not  furnish  another  equation  between  x 
and  y.     We  may  usually  solve  such  a  problem  by  simple  inspection.     Thus,  in 

this  case,  we  have  y  =  — ~       .    Now,  trying  the  integral  values  of  x  till  2x  be- 
3 

comes  greater  than  35,  t.  e.  till  x  =  18,  we  can  determine  what  integral  values 

of  X  give  positive  integral   values   for  y.      For  x=l,    y  =  ll.     For  x  =  2, 

y  =  lOi  ;  hence  a;  =  2  is  to  be  rejected.    For  a?  =  3,  y  =  OJ,  and  a;  =  3  is  to  be 

rejected.    For  a;  =  4,  y  =  9 ;  hence  aj  =  4  and  y  =  9  are  admissible;  etc, 

[Note. — This  subject  is  not  of  sufficient  importance  to  justify  our  going  into 
a  general  discussion  of  it.  We  sliall  content  ourselves  with  a  few  practical 
examples  concerning  simple  indeterminate  equations  between  two  or  three 
quantities,  and  these  restricted  to  positive  integral  solutions.  The  chief  thing  of 
importance  is  t?uit  tlie  student  comprehend  the  nature  of  an  indeterminate  equa- 
tion.l 

*  This  statement  requires  us  to  iuclnde  imaginary  values. 


INDETERMINATE   EQUATIONS.  191 


Examples. 

1.  What  positive,  integral  values  of  x  and  //  will  satisfy  the  equa- 
tion 5:i;+ 7//=  29  ? 

Solution.— We  inuy  write  x  =  — -— ^  =  5—^4-  —~  -—  5— y+  -^--  .    Now 

O  i)  O 

to  make  a;  positive  we  must  have  7y<29;  and  as  y  is  to  be  an  integer  it  can 

2 .. 

only  have  values  less  than  5.     Again,  to  render  x  integral  — — ^  must  be  integral, 

0 

2—v 
or  0.     Finally,  as  no  value  for  y  less  than  5  will  render  — — -  integral  or  0,    ex- 

o 

cept  y=^2,  this  is  the  only  value  of  y  wiiich  fulfills  the  conditions.    Hence  the 

answer  is  y=2,  x='d. 

2.  What  positive,  integral  values  of  x  and  y  will  satisfy  the  equa- 
tion ll:?:-17y=5? 

Solution. — We  have  x  =  — j^ —  =y-r  -^^ — .   From  this  we  see  that  any  pos- 
itive value  of  y  which  will  render  -^ —  integral,  will  meet  the  conditions.    Put 

6y+5  /      .  X        ^       1  llw— 5  ^m—1      _,         i     ^i  .       , 

-^ —  =7U  (an  mteger) ;  whence  y  =  — - —  =m-f  5  — - —  .    To  make  this  value 
11  Go 

of  y  integral  — —  must  be  integral.     Put  — ^p-  =  s  (an  integer) ;    whence  m 

=Qs+l.  Now  any  positive  integral  value  for  s  will  fulfill  the  conditions.  Thus, 
put  8  =  0;*  whence  m=l,  y  —  \,  and  x  =  2.  Again,  put  5  =  1;  whence  m  =  7, 
y=12,  and  a;=19.  For  «  =  2,  m:=13,  y— 23,  and  j^^: 36,  etc.  Hence  there  is  an 
infinite  number  of  positive,  integral  values  of  x  and  y  which  satisfy  the  equation. 

3.  What  positive,  integral  values  of  x  and  y  will  satisfy  the  equa- 
tion 21a: +17^^=2000? 

^     ,              2000-17y              .         ,,^      ,     .           2000-17iy     ^^    5-17y 
SuG's.    x  = — ^  .     .'.  y  is  <  118.     Again  x  = ^ — -  =95h g—  , 

J  5— 17y                       .            ^.            ,                    5— 4m       ,,„  5— Am 

and  — — —  =  m.     .'.  m  is  negative,  and  y=  — m  H p=—  .     Whence  — ^n —  ~  **' 

and    m    =  — z —  =  1  —  4«  H — 7-  .     .*.  «  is  +,  and  any  value  of  »  which  renders 
4  4 

-  — ,  0  or  integral,  and  gives  y  <  118,  will  meet  the  conditions. 

8=    1,  gives  w=—    3,  y=      4  and  a;  =  92. 

8=5,     "     m=  —20,  y=    25  and  a;  =  75. 

*  =   9,     "     m=  —S7,  y=   46  and  x  =  58. 

»  =  13,     "     m=  —54,  y=   67  and  x  —  41. 

»  =  17,     "     m  =  -71,  y=   88  and  x  =  24. 

4  =  21,     "      m=-88,  y=109and.r=    7. 


*  0  is  considered  an  iulcger. 


192  ADVANCED  COURSE  IN  ALGEBRA. 

Since  any  greater  value  of  s  makes  y  >  118,  these  are  all  the  values  of  x  and  y 
which  fulfill  the  conditions. 

4.  Find  the  positive,  integral  values  of  x  and  y  which  satisfy  the 
following : 

(a)  5.r  +     Uy  =     254;  (b)     7a:  +  13?/ =71; 

(c)  dx  +     13y  =  2000;  {d)  ITx  =  542  -  11^; 

(e)  Ux  +    S5y  =    500;  (/)  19:c  -  117.V  =  11; 

(g)  117.T  -  128^  =      95;  (//)  39.r  +  29?/'=  G50; 

(i)  5x  ±       9y  =      40 ;  (k)     6x  ±  Oy  =  37. 


Applications. 

1.  In  how  many  ways  can  I  pay  a  debt  of  $2  with  3-cent  and 
5-cent  pieces  ? 

Sug's, — Let  X  =  the  number  of  3-cent  pieces  and  y  =  the  number  of  5-cent 
pieces  required.  Then  we  are  to  determine  in  how  many  ways  the  equation 
3iF-|-5y=200  can  be  satisfied  for  positive,  integral  values  of  x  and  y. 

We  find  it  to  be  in  13  ways,  as  follows  : 

y  =   11    4  1    7  I  10  I  13  I  16  I  19  I  22  I  25  I  28  I  31  I  34  I  87 
2;  =  65  I  60  I  55  I  50  I  45  I  40  I  35  I  30  I  25  I  20  I  15  I  10  I    5 

This  means  that  1  5-cent  piece  and  65  3-cent  pieces  will  pay  the  debt,  or  4 
5-cent  and  60  3-cent,  or  7  5-cent  and  55  3-cent,  etc. 

2.  A  man  hands  his  grocer  15  and  tells  him  to  put  up  the  worth 
of  it  in  11-cent  and  3-cent  sugars.  Can  the  grocer  do  it  in  even 
pounds?  If  so,  in  how  many  ways?  What  is  the  greatest  number 
of  pounds  of  the  poorer  sugar  that  he  can  use  ?    AVhat  the  least  ? 

3.  In  how  many  ways  can  a  debt  of  £50  be  discharged  with  guineas 
and  3-shilling  pieces?  ^1  ??.<?.,  Not  at  all. 

4.  If  my  creditor  has  only  3-6hilling  pieces  and  I  only  guineas,  can 
he  so  make  change  with  me  that  I  can  pay  liim  £50  ?  Can  I  pay  him 
£201  ?  In  how  many  different  ways  ?  What  is  the  least  number  of 
guineas  and  3-shilling  pieces?  How  is  it  if  I  have  crowns  instead  of 
guineas?  How  if  I  have  guineas  and  my  creditor  crowns  ?  How  if 
I  have  crowns  and  my  creditor  pounds  ? 

5.  In  how  many  ways  can  a  debt  of  £1000  be  paid  in  crowns  and 
guineas  ? 

SuQ. — Having  obtained  a  few  of  the  possible  values  of  x  and  y,  the  law  will 
become  evident. 


indeterminate  equations.  l93 

219.  Indeterminate  Equations  between  Three  Quantities. 

1.  What   are   the   positive  integral  values  of  Xy  y,  and  z  which 
Siitisfy  '6x+by  -h7z=  100  ? 

Solution. — We  have  x  = ~ —  ;  whence  as  1  is  the  least  value  that  y 

or  z  can  have,  x  cannot  be  greater  than  29.    Also  y  =  • '^- ;  whence  y 

cannot  be  greater  than  18.     Also  z= — -  ;   whence  z  cannot  bo    greater 

than  14* 

__,  .^  100-5.y-72     ^.^  ^       l-2p-z      „         l-2v-z 

Write  X  = ^ =33_y_23H ^ —  .    Hence ,j must  be  an 

integer.     Put ^ — -  =m  ;  whence  y=z  —m  -\ .      From  this  we  see 

that  m  is  negative. 

Let  us  now  proceed  to  examine  in  succession  for  2=1,  2=2, 2=3,  etc. 

For  2=1.— For  this  value  of  z,  a;=31— y—  ~ ,  and  y=  —  m  —  — .     From  the 

o  -* 

latter  we  see  that  m  must  be  an  even  negative  number ;  and  from  the  former, 

that  y  must  be  a  multiple  of  3.     Hence  the  following  computation : 

For  m=        0,  1/  =   0,  which  is  inadmissible. 

For  w  =  -    2,  y  =    3,  and  x  -  26. 

For  m—  —   4,  y  —   C,  and  a;  =  21. 

For  m  =  —   6,  y=   9,  and  a;  =  16. 

For  m  =  —   8,  y  =  12,  and  x  =  11. 

For  wi  =  —  10,  y  —  15,  and  x  —    6. 

For  ?»  =  —  12,  y  =  18,  and  a;  =    1. 

Since  the  values  of  x  decrease  as  m  increases  numerically,  and  1  is  the  least 
admissible  value  of  x,  we  have  all  the  values  of  y  and  x  which  correspond  to 
«  =  1. 

For  2  =  2.— For    this    value   of    z,  a;  =  28  —  y  4-  -^^—  »    *^<1    y——m 

—  .      From  the  latter  we  see  that  m  must  be  a  negative  oM  number  ; 

i 
and  from  the  former,  that  y  must  be  1,  or  a  unit  more  than  a  multiple  of  3. 

Hence  the  following  computation  : 

For  ?7i  =  —    1 ,  y  —    1 ,  and  a;  =  27. 

For  m  =  —   3,  y  —    4,  and  x  —  22. 

For  m—  —    5,  y  =    7,  and  x  —  17, 

For  w  =  —    7,  y  =  10,  and  x  =  12. 

For  m  =  —    9,  y  =  13,  and  a;  =    7. 

For  m  =  —  11,  y  =  16,  and  x  =    2. 
Henco,  these  are  all  the  values  of  y  and  x  wliich  correspond  tq  2  =  2, 


Of  coarse  the  quantities  uecd  not  come  up  to  ^ese  limits. 


194                                ADVANCED   COURSE  IN   ALGEBRA. 
The  other  values  are  as  follows : 

'-**  U=23  I  18  I  la  I    81    8  I  •  "'*  U=19  I  14  I  9  I    4 

.  (y=  1  I    41    7  1  101  „  iy=  2  1    5  I  8  I  11 

'='u=2oll5llol    sl'  ^=«U=16lll     6       1 


:5 

2.  What  positive,  integral  values  of  x,  y,  and  z  satisfy  17a;  +  lOjf 
+  21;?  =  400? 

SuG. — There  are  10  sets  of  values. 

3.  AVhat  positive,  integral  values  of  x,  y,  and  z  satisfy  5x  +  "71/ 
+  11;?  =  224? 

4.  What  positive,  integral  values  of  a;,  y,  and  z  satisfy  Gx  +  Sy 
+  6z  =  12  ?     Also  2a;  +  3//  +  52  =  41  ? 


220,  If  tlic  conditions  of  a  problem  furnish  less  equations  than 
unknown  quantities,  the  problem  is  mdetermmate,  and  in  general 
can  have  an  infinite  number  of  solutions.  But  if  the  solution  be 
limited  to  positive,  integral  values,  it  can  be  effected  as  above.  Thus, 
if  there  are  two  equations  and  three  unknown  quantities,  one  of 
the  unknown  quantities  can  be  eliminated  and  the  resulting  equation 
solved  as  heretofore.  In  like  manner  if  there  are  three  equations 
and  four  unknown  quantities,  a  single  equation  between  two  may  be 
found  and  solved ;  or  if  four  unknown  quantities  and  but  two  equa- 
tions, a  single  equation  between  three  unknown  quantities  may  be 
found  and  solved. 

Examples. 

1.  Given  2a;  +  5y  +  3^  =  51,  and  10a;  +  3^  +  2^  =  120,  to  find  all 
the  positive,  integral  values  of  a;,  y,  and  z. 

2.  Given  3.?;  +  5?/  +  7z  =  5G0,  and  \)x  +  25?/  +  495;  =  2920,  to  find 
all  the  positive,  integral  values  of  x,  y,  and  z. 

3.  Given  2a;  4-  H//  -Zz=  10,  and  3a;  -  2y  +  3z  =  30,  to  find  all 
the  i)ositive,  integral  values  of  a*,  y,  and  z. 


INDETERMINATE   EQUATIONS.  19^ 


Application's. 

1.  I  wish  to  expend  $100  in  the  purchase  of  three  grades  of  sheep, 
worth  respectively  $3, 17,  and  $17  per  head.  How  many  of  each  kind 
can  I  buy  ?  In  how  many  different  ways  can  I  make  the  purchase  ? 
How  many  of  the  first  two  kinds  must  I  take  in  order  to  get  the  least 
possible  number  of  the  third  kind  ? 

2.  A  merchant  has  three  kinds  of  goods.  The  value  of  20  yards 
of  the  first,  less  the  value  of  21  yards  of  the  second,  is  $38 ;  while 
the  value  of  3  yards  of  the  second  and  4  yards  of  the  third  is  $34. 
What  is  the  price  per  yard  of  each  kind,  the  question  being  restricted 
to  even  dollars  ?     AVhat  if  the  latter  restriction  be  removed  ? 

3.  In  how  many  ways  can  I  pay  a  debt  of  $171  with  $20,  $15,  and 
$6  notes?  What  is  the  least  number  of  $20  notes  that  I  can  use? 
Of  $15  notes  ?     What  the  greatest  number  of  $6  notes  ? 

4.  A  farmer  has  calves  worth  $10,  $11,  and  $13  per  head.  What 
relative  number  of  each  must  he  take  and  sell  them  at  the  uniform 
rate  of  $12,  without  gain  or  loss  ?  If  he  is  to  sell  only  15  animals, 
how  must  ]ie  select  them  ? 

5.  A  man  bouglit  124  head  of  cattle,  viz.,  pigs,  goats,  and  sheep, 
for  $400.  Each  pig  cost  $4^,  each  goat  $3|,  and  each  sheep  $1^. 
How  many  were  there  of  each  kind  ? 

G.  A  grocer  has  an  order  for  150  pounds  of  tea  at  90  cents  per  pound, 
but  having  none  at  that  price,  he  would  mix  some  at  75  cents,  some 
at  87^^  cents,  and  some  at  $1.00  per  pound.  How  much  of  each  sort 
must  he  take  ? 

Sug's.— The  nature  of  the  4th  and  5tli  problems  restricts  their  solutions  to 
positive  integers.  The  6th  is,  however,  only  restricted  by  its  nature  to  positive 
numbers ;  they  may  be  fractional  as  well  as  integral. 

[See  Complete  School  Algebra,  subject  Alligation.] 

7.  What  quantity  of  raisins,  at  10  cents,  18  cents,  and  20  cents  per 
pound,  must  be  mixed  together  to  fill  a  cask  containing  150  pounds, 
and  to  be  worth  .19  cents  a  pound  ? 

8.  A  wheel  in  36  revolutions  passes  over  29  yards ;  and  in  x  of 
these  revolutions  it  describes  z  yds.,  y  ft,  and  5  in.  What  are  the 
values  of  x,  ?/,  and  z  ? 


196 


ADVANCED  COURSE  IN  ALGEBRA. 


CHAPTER  IL 


LOCI   OF    EQUATIONS. 

[Note. — This  subject,  though  properly  geometrical,  is  introduced  here  for  the 
purpose  of  the  elegant  and  clear  illustrations  which  it  affords  of  the  abstract 
principles  of  the  subject  of  Higher  Equations.  It  is  thought  that  the  aid  which 
it  will  afford  the  pupil  in  comprehending  the  principles  of  the  succeeding  chapter 
will  more  than  compensate  for  the  time  required  to  master  this.  Moreover,  the 
subject  of  Loci  of  Elquations  is  of  prime  importance  in  a  mathematical  course, 
and  is  always  pursued  with  pleasure  by  the  pupil.  No  geometrical  knowledge 
is  required  in  reading  this  chapter,  farther  than  the  ability  to  draw  and  measure 
straight  lines.] 

22 !•  JPvop» — Every  equation  between  two  variables,'^  having 
real  root8^\  may  be  interpreted  as  representing  some  line  either 
straight  or  curved. 

This  proposition  will  be  made  sufficiently  evident  for  our  present  purpose,  if 
we  show  how  such  equations  can  be  made  to  represent  lines.  This  we  shall  do 
by  means  of  particular  examples. 

Examples. 

1.  Draw  the  line  represented  by  the  equa- 
tion 1/  =  2x  ■{■  6. 

Solution. — First,  in  all  cases,  draw  two  straight 
lines,  as  X'X  and  YY'  ,at  right  angles  to  each  other, 
as  in  the  figure.  Then, in  the  equation  y  =  2x  +  Q, 
assign  values  (arbitrarily)  to  x,  and  find  the  corre- 
sponding  values  of  j/.    Thus, 

Also,  if  x=—l, 

"     x=-2, 
"     x=-Z, 


-5-4     h 


■»-t-iAI 


If  a;=0, 

y=  6. 

"  x=l. 

P=  8, 

"  x=2, 

y=io. 

"   ar=3. 

y=i2, 

"   x=4, 

y=i4. 

etc.. 

etc. 

x=-4, 
x=—5, 
etc.. 


y=  4, 
y=  2, 
y=   0. 

y=-4, 
etc. 


Y 

Fio.  1. 


Having  computed  a  few  corresponding  values 
of  X  and  y  in  this  way,  we  proceed  with  the  figure, 
as  follows :  Measure  off  a  distance  Al  to  the  light 


*  This  means  eimply,  "  having  two  variables,  and  only  two,  in  it." 

t  The  geometrical  interpretation  of   imaginary  loci  does  not  come  within  our   present 
purpose. 


LOCI   OF  EQUATIONS. 


197 


of  A,  of  some  convenient  length,  and  call  it  the  unit  of  distances.     Draw  6 1, 

at  1,  perpendicular  to  AX,  and  make  it  8  units  long  (t.  e.,  8  times  as  long  as  Al). 

Now,  6  is  at  a  distance  1  to  the  right  of  the  line  YY'.  and  8  above  the  line  X'X, 

and  is  hence  a  point  in  the  line  which  our  equation  represents.    In  like  manner, 

find  the  point  c,  2  to  the  right  of  YY',  and  10  above 

X'X  ;  and  c  is  another  point  in  the  line  represented 

by  our    equation.      Again,   when  a;  =  3,    y  =  13. 

Hence,  lay  off  three  units  to  the  right,  as  to  3  in  the 

figure,  and  draw  d  3  perpendicular  to  X'X  and  12  in 

length.     Then  is  d  another  point   in  the  line  we 

seek.    When  a;  =  4,  y  =  14.    Hence  e  is  a  point  in 

the  line  ;  since  it  is  4  from  YY',  and  14  from  X'X 

When  a;  =  0,  y  =  6  ;    whence  «  is  a  point  in  the 

line,  as  it  is  0  distance  from  YY',  and  6  from  X'X. 

For  negative  values  of  a;,  we  have,  when  x  =z  —  1, 
y  =  4.     Now,  laying  off  negative  values  of  x  to  the 
left  from  A,  since  we  laid  off  positive  values  to  the 
right,  we  measure  from  A  to  —1,  the  unit's  distance, 
take  /I  equal  to  4  units,  and  thus  find  the  point  /. 
When  X  =  —  2,  y  =  2,  and  g  is  the  corresponding 
point.     When  a;  =  —  3,  y  =  0  ;  whence  h  is  a  point 
in  the  line,  as  it  is  3  to  the  left  of  YY'  and  0  above 
X'X.     When  a;  =  —  4,  y  =  —  2.     As  this  value  of  y 
is  negative,  we  lay  it  off  below  X'X.    Thus,  taking 
from  A  to  —4,  a  distance  of  4  units,  and  from  —  4  to  e,  a  distance  of  2  units,  *  is 
a  point  in  the  line.     Thus  also  A:  is  a  point  in  the  line,  since   when  a;  =  —  5, 
y  =  —  4,  and  k  is  taken  5  to  the  left  of  YY'  and  4 
below  X'X. 

This  process  might  be  continued  indefinitely, 
both  for  y)ositive  and  negative  values  of  x.  We 
might  also  use  fractional  values  of  a*,  as  a;  =  \, 
x  =  ^,x  —  2^,  etc.,  and,  finding  the  corresponding 
values  of  y,  locate  points  between  those  found  by 
taking  integral  values. 

Finally,  joining  the  points  e,  d,  c,  b,  a,f,  g,  h,  i, 
k,  we  have  the  line  MN,  which  is  represented  by 
the  equation  y=:2x-{-Q.  This  line  does  not  stop 
at  M  and  N,  of  course,  since  we  might  produce 
it  indefinitely  either  way,  by  continuing  to  take 
larger  and  larger  values  of  x  (numerically).  In 
this  case  it  is  easy  to  see  that  the  line  is  an  indefi- 
nite straight  line. 

2.  What  line  is  represented  by  the  eqiia-  f^°-  s. 

tiony=Sx-Q?     (^ee  Fir/.d.) 

SuG's.— First  compute  a  table  of  corresponding  values  of  x  and  y,  as  in  the 
preceding  example ;  and  then  locate  the  points  thus  designated. 


108 


ADVANCED  COURSE  IN  ALGEBRA. 


3.  What  line  is  represented  by  the  equation  y  =  —2a; +  4?     (See 
Fig.  4.) 

222.  Definitions.— The  assumed  fixed 
lines  X'X  and  YY'  are  called  the  Axes  of 
Meferencey  or  simply  the  Axes.  A  is 
called  the  Origin.  X'X  is  the  Axis  of  Ab- 
scissas, and  YY'  the  Ax'is  of  Ordinates. 
The  distance  of  a  point  from  the  axis  of  ab- 
scissas is  called  the  Ordinate  of  the  point ; 
and  the  distance  of  a  point  from  the  axis  of 
ordinates  is  called  its  Abscissa.  The  ordi- 
nate and  abscissa  of  a  p)int  taken  together 
are  called  its  Co-ordinates. 

Abscissas  measured  to  the  right  from  the  axis  of  ordinates  are  +, 
and  those  to  the  left  — .  Ordinates  measured  above  the  axis  of 
abscissas  are  +,  and  those  below  — . 

4  to  13.  Draw  as  above  the  lines  represented  by  the  following  equa- 
tions:   i/=x-\-5;    y=x—5;    y=  — .t  +  5;     y=  —x—5;    t/=ix-{-(j; 

y=4a;-6;    y= -4x  +  6;    y=  -ix-G;    2a;-3?/=-5;^i^  =2. 

Sua. — Put  such  equations  as  the  last  two  into  the  same  form  as  the  others 
before  proceeding  with  the  solution  as  above. 


OiOf 


V,  Def. — The  line  which  is  represented  by  an  equation  is 
called  the  Locus  of  the  equation;  and  drawing  the  line  in  the 
manner  indicated,  is  called  Constrtictbiy  ilva  Locus  of  the  equation. 

X 


14.  Construct  the  locus  of  the  equation  y  = 


l+a'^' 


Solution.— For  x  =  0,    y  =  0, 


"    x  =  i, 

y  =  -1*7, 

"    x=A, 

y  =  h 

*'    x  =  l. 

y  =  h 

"      X=:2, 

y  =  h 

"    x  =  d, 

y=h. 

"    x  =  A, 

y  =  A. 

etc. 

etc. 

For  a;  =  -  i.  y  =  -  %, 

"    x=  -1,  y=  -h 

"    x=  -2,  y=  -h 

"    .-c  -  -  3,  y=  —  -,%, 

"    x=  -A,  y=  —  -iV» 
etc.  etc. 


LOCI   OF   EQUATIONS. 


199 


Now,  laying  off  on  the  axis  of  abscissas  to  the  right  distances  equal  to  \,  J, 
1,  2,  3,  and  4,  on  some  convenient  scale,  and  at  these  points  erecting  ordinates 
equal  respectively  to  -,\,  |,  i,  f,  ,%,  and  A  of  the  same  scale,  we  find  the  points  a, 
b,  c,  d,  e,  and  /  of  the  locus.  We  also  see  that  if  we  continued  to  give  x  greater 
and  greater  values,  y  would  continually  grow  less,  but  would  only  become  0  when 

«  =  00,  for  then  we  should  have  y  — =  —  —-—  —  =  0. 

In  like  manner  laying  off  the  negative  values  of  a;,  and  the  corresponding 
values  of  y,  we  find  the  points  a\  b' ,  c' ,  d',  e',  and/',  and  also  find  that  y  dimin- 
ishes numerically  as  x  increases  numerically,  and  that  for  x  negative  y  is 
always  negativ^e,  and  only  becomes  0  when  x=  —  oo.  Hence,  the  curve  ap- 
proaches the  axis  of  abscissas  to  the  left  from  below,  as  it  does  to  the  right 
from  above,  reaching  it  in  either  direction  only  at  an  infinite  distance  from  the 
origin. 

A  line  sketched  through  the  points  found  represents  the  locus  sought. 


15  to  18.  Construct  the  loci  of  the  following  equations:  i/=x^ 
+  a:--6t  (see  Fig.  6) ;  y=3-{-x—ix^  (see  Fir/.  7);  i/=x'—4:X  +  4: 
(see  Fig,  8) ;  g=x^—3x-{-5  (see  Fig.  9). 

Y 
N|  ,M 


ii\ 


nr, 


X'    -2-W   12346         X 


Y 

Fig.  9. 


•  Dropping  the  finite  quantity  1,  as  prodiicinj;:  no  effect  when  added  to  the  infinite  ar*. 

t  In  dctorininin-j  points  in  tlio  Iociib.  it  is  often  neceeeary  to  attribute  fractional  valiics  to  x. 
Thue,  in  tiiis  cnso.  to  sketch  the  curve  from  a  to  c\  we  need  an  intermediate  point.  If  (here  is 
any  doubt  about  the  character  of  the  curve  between  two  points,  resolve  the  doubt  in  thi:*  way. 


200 


ADVANCED  COURSE  IN  ALGEBRA. 


19  to  23.  Consfcrnct  the  loci  of  the  following  equations:  %j=y* 
-Ja;'  +  2aj-f  2  (see  Fig.  10)  ;  y-^-(^j? 
+  13ar-10  (see  Fig.  11) ;  y=x'-2x-5  (see 
Fig.  12);  y=a^(o-x)  (see  Fig.  13);  and 
y=o^-Q:xf-^llx-6  (see  F'ig.  14). 


Fie.  16. 


loci   OF  EQUATIONS. 


201 


24  to  28.  Construct  the  loci  of  the  following  equations:  y^a"^ 
~5ic2+4  (see  Fig.  15);  yz=x'-\-'23?-'6x'-ix  +  4.  (see  i^7^.'l6); 
y=zx'-^x'  +  ^x-^n  (see  Fig.  17);  y=i}^-2a^-7x'-Sxi-l()  (see 
Fig.  18)  ;  and  g=a^+a^+x'-{-x-\-l  (see  Fig.  19). 


X'      A 


X'    » 


'      X 


Y 

Fig.  18. 


29.  Construct  the  locus  of  the  equation  i/=x^  +  4:X*—14:X^—17x—6. 

Sug's.  —  In  attempting  to  construct  this 
locus,  it  is  necessary  to  give  x  values  from 
—3  to  +2,  including  these  values,  and  also 
to  observe  the  character  of  the  locus  beyond 
these  limits.  But  it  will  be  found  that  for 
some  values  of  x  between  these  limits, y  is  in- 
conveniently large.  In  sketching  the  figure, 
we  may  use  one  scale  for  laying  off  values  of 
X,  and  another  for  laying  off  values  of  y. 
Thus  in  the  figure  given,  the  unit  used  for  x 
is  6  times  as  great  as  that  used  for  y.  This 
is  equivalent  to  constructing  the  locus  Gy=x'^ 
+  4x*  -  Ux^-  llx  -  6,  or  y  =  ^x'  +  W-W 
—  Va;  —  1.  This  locus  has  all  the  peculiar- 
ities of  the  one  required  (that  is,  all  the  turns, 
flexures,  or  bends),  but  is  not  of  the  same  pro- 
portions. The  portion  represented  is  G  times 
as  wide  in  relation  to  its  length  as  the  re- 
quired locus  would  have  been. 

30  to  41.  Construct  the  loci  of  the    'y. 
following  equations,   using   a   smaller 
scale  for  /y  than  for  x,  as  explained  in 


ADVANCED   COURSE   IN   ALGEBllA. 


the  suggestions  above,  when  more  convenient :  y  =  x^—  2x  —  15 ; 
y  =  x'  +  2x  +  2;  y  =  x''-10x+26'y  y  =  x' -  3x'  -  Hx- 10; 
y  =  23?-  12a:»  +  13a;  -  15 ;  y  =  x"  -  x"" - '^x  ■{■  12 ;  y  =  o?  -  2x^ 
-  25a:  +  50;  y=  ir*—  2r'+  8a;  -  16  ;  y=  a;'+  2%''—  Zx"-  4a;  +  4; 
y  =  a;*-  6a;^  +  5a;'  +  2a;  -  10 ;  y  =  ar^  +  5a;^  +  a;^-  16a;''-  20a;  -  16  ; 
and    y  =  5a:'-4a;^  +  3a;'-3a;'+4a;-5. 


224,  J^vob, — To  construct  the  real  roots  of  an  equation  contain- 
ing only  one  unknoio}i  quantity. 

Solution. — Put  the  equation  in  the  form  /(a)  =  0,  tlien  write  y  =  f{x). 
Construct  this  equation,  and  the  abscissas  of  the  points  where  the  locus  cuts  the 
axis  of  abscissas  are  the  nwts  of  the  equation  /(.r)  =  0.  This  is  evident,  since 
for  these  points,  and  for  these  only,  y  =  0,  and  we  have  f{x)  =  0. 


Examples. 
1.  Construct  the  real  roots  of  the  equation 


3a;  -2  =  0. 


Solution.— We  will  first  write  y  z=zx*—^  —  2 

for  x=\,  y  - 


X' 


Fio.  21. 


Now,  for  a;  =  0,  y  —  —  2\ 
—  4  ;  for  x  —  2,  y  =  —  4  ; 
for  a;  =  3,  y  —  —  ^\  and  for  a;  =  4,  y=2. 
Hence  we  see  that  the  locus  of  the  equa- 
tion y  =  ic*  —  3x  —  2,  cuts  the  axis  of  ab- 
scissas between  a*  =  3,  and  x  =  4,  since  it 
passes  from  below  the  axis  of  abscissas 
(where  y  is  — )  to  above  this  axis  (where 
y  is  -h).  There  is  therefore  a  root  of  a;* 
—  3a:  —  2  =  0  between  3  and  4.  To  con- 
struct this  root,  we  sketch  the  curve  be- 
tween a*  —  3  and  x  =  4,  by  finding  the 
values  of  y  for  a  few  intermediate  values 
of  .T,  and  then  sketching  the  curve.  Thus 
for  a;  =  3^,  y  =  —  J4  ;  for  a*  =  35,  y  =  \h  Sketching  the  curve  mil  through 
these  points,  we  find  by  measurement  Art  =  3.50,  as  an  api)roximate  value  of  x 
in  the  equation  a;*—  3.r  —  2  —  0.  (Verify  by  solving  the  equation.)  To  construct 
the  other  root,  we  notice  that*  for  x  =  0,  y  =  —  2,  and  the  curve  cuts  the  axis  of 
abscissas  again  at  the  left  of  the  origin  (probably,  as  it  certainly  does  not  cut  it 
again  at  the  right).  Now,  for  x  =  —  1,  y  =2 ;  whence  we  see  that  the  locus 
cuts  the  axis  between  a;  =  0,  and  x  =  —  1.  For  x——i,y=—\;  and  for 
x=  —  },  y  =  If .  Sketching  the  curve  through  these  points,  we  have  m'n' ; 
and  measuring  A  a',  we  find  the  other  value  of  x  to  be  —.56. 

SuG, — For  constructing  the  approximate  roots  in  this  manner,  as  we  only 
need  to  sketch  a  small  portion  of  the  locus,  in  the  vicinity  of  its  intersection  with 
the  axis,  we  can  use  a  much  larger  scale  than  would  otherwise  be  practicable, 
and  thus  obtain  a  nearer  approximation.     With  good  instruments  and  some  care, 


NUMERICAL   HIGHEH   EQUATIO:s'S.  203 

we  can  usually  construct  the  root  with  tolerable  accuracy  to  hundredths.  When 
the  locus  cuts  the  axis  quite  obliquely,  the  approximation  cannot  be  made  as 
accurate. 

2  to  7.  As  above,  construct  the  real  roots  of  the  following: 
x^-8x=U',  a^-  12a:'-'  +  36:c  -  7  =  0 ;  a:^  -  x'  -  lOx  +  6  =  0  ; 
a^  -  7a:  +  7  =  0 ;  x*  -  12a:3  +  60x'  -  84a:  +  49  =  0;  and 

2x'  -  7x'  +  10a:  =  9. 

225,  Sen. — This  method  of  approximating  the  roots  of  equations  geo- 
metrically is  not  given  as  a  good  practical  method ;  but  simply  to  assist  tha 
learner  in  comprehending  some  subsequent  processes,  and  for  its  geometrical 
importance. 


CHAPTER  III. 

HIGHER  EQUATIONS. 


SECTION  I. 

SOLUTION  OF  NUMERICAL  HIGHER  EQUATIONS  HAVING  COMMEN. 
SURABLE  (OR  RATIONAL)  ROOTS.* 

226 »  Equations  of  higher  degrees  than  the  second  are  called 
Higher  Equations  {6-10 ,  or  same  in  Complete  School  Algebra). 
Ko  general,  practicable  method  of  resolving  such  equations  is  known. 
Theoretical  solutions  of  equations  of  the  third  and  fourth  degrees 
(cubics  and  biquadratics)  are  known ;  but  these  solutions  are 
attended  with  practical  difficulties  in  many  cases,  which  render 
them  nearly  or  quite  useless.  We  are,  however,  able  to  obtain  the 
real  roots  of  Niimerical  Higher  Equations,  in  all  cases,  either  exactly, 
or  to  any  required  degree  of  approximate  accuracy. 

227,  The  Real,  Covimensnrable  Roots  of  numerical  equations  are 
usually  found  with  little  difficulty  by  the  methods  given  in  this 
section. 

*  A  commensurable  root  (or  number)  is  one  which  can  be  exactly  expressed  in  the  decimal 
notation,  either  in  an  integral,  fractional,  or  mixed  form.  Thus,  4,  ^l,  r25,  etc.,  are  com- 
mcnvurable.    But    V2\,    VXH,  etc.,  arc  incomirenpurablc. 


204  ADVANCED   COURSE   IN  ALGEBRA. 

228,  I^VOJ),  —  fJcery  equation  with  oite  nnknoicn  quantity^ 
having  real  and  rational  coefficients^  cafi  be  transfortned  into  an 
equatio)i  of  the  form 

x"+  Ax"-'-f  Bx"-'+  Cx"-^'    -    -    -    -     L  =  0, 

in  which  the  exponents  are  all  positive  inteyers^  the  coefficient  of  x"  is 
1,  and  the  coefficients  of  the  other  terms^  A,  B,  C,  etc.,  and  also  the 
absolute  term  L,  are  integers. 

Dem. — 1st,  If  the  unknown  quantity  occurs  in  any  of  the  denominators  in  the 
given  equation,  remove  it  to  the  numerator  by  clearing  of  fractious.  If  there 
are  tlien  any  negative  exponents,  multiply  each  term  by  the  unknown  quantity 
with  a  i>ositive  exponent  equal  to  the  numerically  largest  negative  exponent. 
Then  unite  the  terms  with  reference  to  the  unknown  quantity,  and  write  them 
in  order  with  the  term  containing  the  highest  exponent,  at  the  left,  and  so  that 
the  exponents  shall  diminish  toward  the  right,  transposing  all  the  terms  to  the 
first  member.    The  most  complicated  form  which  can  then  occur  is 

m  r 

^y"  +^y'  +yy'  -  -  •  •  i  =  o,  (i) 

Tfi  r 
in  which  any  or  all  of  the  exponents  may  be  fractions ;    and  —>-><,  etc. 

n  8 
is  supposed. 

2d.  To  free  the  equation  of  fractional  exponents,  substitute  for  the  unknown 
quantity  a  new  unknown  quantity  with  an  exponent  equal  to  the  least  common 
multiple  of  the  denominators  of  the  exponents  in  the  equation.     Thus,  in  (1) 

m  r 

put  y  —  z"' ,  whence  y  "  =  af",  y '  =  z"',  and  y'  =  2"*'.  These  values  substituted 
in  the  equation,  will  evidently  give  an  equation  of  the  form 

!^r' +  -«"->+ 42*-'''    -    -    -    -    ^  =  0,  (2) 

0(1  J 

in  which  all  that  is  essential  concerning  the  exponents  is  that  they  should  be 

all  positive  integers,  decreasing    in  value   from    left    to  right,  since  iji  (1) 

m       r 

—  >  -  >  ^  etc. 

n      H 

3d,  Now  divide  by  the  coefficient  of  z*,  and  let  the  resulting  equation  be 
represented  by 

2"+ ^,2-' +-6--^    .    .    .    .    ;'=0.  (3) 

d  f 

Finally,  put  z  =  —  ,  and  substitute  in  (3),  thus  obtaining 

r-^^nF---'^fi^^  ■  ■  ■  ■  '  =  »•  (*) 

Multiplying  (4)  by  k",  and  representing  the  absolute  term  by  L,  we  hare 


NUMERICAL   HIGHER   EQUATIONS.  205 

If  now  k  be  so  taken  that  these  numerators  will  be  divisible  by  the  denomina- 
tors, and  the  quotients  represented  b^  A,B,  C,  etc.,  we  have 

a;»4-^a;~-i-|-i?a;'»-2-f  Caj^-a    ....    Z=0, 
the  form  required. 


Examples. 

1  1  2  K  Q 

1.  Transform  -  +  ^x-'  +  ^x^  =  2x^  +  iz~^  i-~-2,  into  a  form 

having  positive  integral  exponents  and  coefficients,  and  having  the 
coefficient  of  the  highest  power  1. 

Solution. — Multiplying  by  x^,  we  have 

x+%v-^-\-^x^=2J-hhic~^+d-2xK  (1) 

Multiplying  (1)  by  x"^,  we  have 

x*  +  i+^^x^^z=2x^+{x^+3x'-2j*.  (2) 

Putting  x=y^ ,  there  results 

y''+h+^f'=^'*+\y''^^t'-2f\ 

Arranging  with  reference  to  the  highest  power  of  y, 

23/"-^y"-2/"-/*+3y"'+iy"-J=0,  or 

z 
Finally,  put  y  =  -r,  whence 

Now,  if  A;  be  made  12,  this  equation  will  be  of  the  required  form  * 

Z  2* 

Notice  that  as  ic  =  y',  and  y  —  :r^,x  =  -r^  ;  so  that,  if  the  value  of  z  could 
be  found,  the  value  of  x  would  be  known  by  implication. 

2.  Show  as  above  how  to  transform  the  following: 

{a)  Sy-"^  +  iy-i  +  ^  -  if^  =  ^  +  if-  ^Vi 
(b)  --Zx  +  ix^-l  =  l; 

X 

(e)   Vl-ar'=l~3a:i;  (/)  ^2^ 3? -- a:  =  a/T^. 

♦  This  substltntion  would  be  tedious,  and  as  it  is  our  present  purpose  simply  to  show  the 
poefibUify  of  the  traneformation,  and  the  method  of  making  it,  the  enbetitutlon  1»  iinnfce*fary. 


206  ADVANCED  COURSE  IX  ALGEBRA. 

^*?.9. — Since  every  equjiiion  with  one  im known  quantity,  and  real 
and  rational  coefficients,  can  be  transibrmed  into  one  of  tlie  Ibrm 

ar-^Ax^-'  +  Bx^-'-j-Cx"^-' L=0,  (I) 

this  will  be  taken  as  the  typical  numerical  equation  whose  solution 
we  sliall  seek  in  this  and  the  succeeding  sections;  and  we  shall 
frequently  represent  it  by/(.r)  =  0,  read  *'  function  x  equals  0."  The 
notation/(a:)  signifies  in  general,  as  has  been  before  explained,  simply 
any  expression  involving  x.  Here  we  use  it  for  this  particular  form 
of  expression.  We  shall  also  use  f'{x)  as  the  symbol  for  the  first 
differential  coefficient  of  this  function. 


230,  J*V02>» — W/t€7i   an   efjuafion    is  reduced  to  the  form  x" 

4-  Ax"-'  -\-  Bx-'  +  Cx°-' L  =  0,  t/ie  roots,  with  their  slgiis 

changed,  are  factors  of  the  absolute  {knotcn)  term,  L. 

Dkm. — 1st.  The  equation  Ix-inrr  in  this  form,  it  a  is  a  root,  the  function  is 
divisible  by  ar—a.  For,  sui)Jkks«.'  ui>on  trial  x—n  goes  into  the  polynomial  xn 
+  J.r"-'  +  ,  etc.,  Q  times  \cith  a  remainder  11.  {Q  represents  any  series  of  terms 
wliicli  may  arise  from  such  a  division,  and  JR  any  remainder.)  Now,  since  the 
quotient  multiplied  by  the  divisor  +  the  remainder,  equals  the  dividend,  we 

liave  (t— a)  Q  +  R=je*  +  .4.r"->  +  2?.i'"-'^  +  Cx'-^ L.     But  this  polynomial  =  0. 

Hence  {x—a)  Q  +  R—O.     Now,  by  hyjwthesis  a  is  a  root,  and  consequently  a;— a 
=0.    Whence  B=0,  or  there  is  no  remainder. 

2d.  If  now  z—a  exactly  divides  x"  +  A.r"-'^  +  Bx'*-^  +  Cz"-^ L,  a  must 

exactly  divide  Z,  as  readily  appears  from  considering-  the  process  of  division. 
Hence  —a  is  a  factor  of  L,  a  being  a  root  of  the  equation,     q.  e.  d. 

23 !•  Cor.  1.— T/'a  is  a  root  o/f(x)  =  0,  f(x)  is  divisible  by  x— a/ 
and,  converselt/j  if  f(x)  is  divisible  by  x  — a,  a  is  a  root  of  f(x)=0. 

Dem. — The  first  statement  is  demonstrated  in  the  proposition,  and  the  second 
is  evident,  since  as^a*)  is  divisible  by;r— r/,  let  the  quotient  be  q>{x)\  whence 
(r— «)  ^.r)=0.  Now  x—a  will  satisfy  this  equation,  since  it  renders  a*— a=0, 
and  does  not  render  <p{x)  infinity,  since  by  hyi)othesis  x  docs  not  occur  in  the 
denominator.* 


232 •  JProj). — If  the  coefficients  and  absolute  term  in  x""  +  Ax""^ 
-f  Bx"-'-f  Cx""' -  -  -  -  L=0,  are  all  inteycrs,  the  equation  can  leave 
it  o  fractional  root. 

*  Could  thert'  be  n  term  of  the  form in  <?>  (r),  x=a  would  render  it  oo,  and  (x-ay^P  (a-)  would 

bv  0  X  ae.  which  is  iiuleterminatc.  since  0  x«=Ox  ^^J}. 


NUMERICAL   HIGHER    EQUATIONS.  •  207 

Dem.— Suppose  in  tliis  equation  x  =  p  ^  being  a  simple  fraction  in  its  lowest 
terms.     Substituting  this  value  of  (v,  we  have 

gn  ,^.n-l  *"-2  on- 3 

Multiplying  by  <"-'  we  obtain 

Now,  by  hypothesis,  all  the  terms  except  the  first  are  integral,  and  the  fiist  is  a 
simple  fraction  in  its  lowest  terms,  as  by  hypothesis  s  and  t  are  prime  to  eacli 
other.  But  the  sum  of  a  simple  fraction  in  its  lowest  terms  and  a  series  of  in- 
tegers cannot  be  0.     Therefore  x  cannot  equal  -,  a  fraction. 

t 

233,  Scu. — Tliis  proposition  does  not   preclude  the  possibility  of  surd 
roots  in  this  form  of  equation.     These  are  possible. 


234.  ProiK—An  equation  f  (x)  =  0  (229)  of  the  nth  degree, 
has  n  roots  {if  it  has  any  *),  and  no  more. 

Dem. — Let  rt  be  a  root  of  f{x)  =  0,  which  is  of  the  7ith  degree.  Dividing 
f(x)  hy  X  —  a  {231),  we  have  <p(j)  =  0,  an  equation  of  the  (/i— l)th  degree. 

Let  5  be  a  root  of  <^.r)  =  0,and  divide  <p(.r)  by  x—h  {231).  Call  the  quotient 
<p' {x),  whence  cp'  {x)  =  0,  an  equation  of  the  («— 2)th  degree.  In  this  way  the 
degree  of  the  equation  can  be  diminished  by  division  until,  after  n  —  1  divisions, 
there  results  (p"  {x)  f  of  the  first  degree,  and  the  equation  is  x—  l—O.     Therefore,. 

f{x)  =  {x  -  a)  (p(,v)  -  {x-a)  {x  -  b)  cp'  {.i)  =  {.v  -  a)  {x  -  h)  {x  -  c)  q>"  {x) 

=  {x-  a)  {x-h){x  -  c) (^  -  ^)  -  0  ; 

%.  e.,  f{x)  is  resolvable  into  n  factors,  of  the  form  x  —  m. 


*  Wc  shall  assume  that  every  equation  has  a  root  real  or  imasjinary ;  i.  e.,  that  there  is  some 
form  of  exprcpgion  which  substituted  for  the  unknown  quantity  will  satisfy  the  equation.  It 
is  shown  in  works  treating  more  largely  upon  the  theory  of  equations,  that  the  general  form  of 
a  root  is  a  +  /?  |/  — 1.  When  /^  =  0,  the  root  is  real.  The  general  demonstration  of  this  propo- 
sition is  loo  abstnisc  for  an  elementary  treatise.  That  every  equation  of  the  form  a-"+  Ax^'-^ 
+  Bx"^~'i+  Cz"-3  -  -  -  -  Z=0  {229)  has  a  real  root  when  n  is  an  odd  number,  and  also 
when  n  is  an  even  number  if  L  be  negative,  is  very  simidc.  Thus,  if  n  is  odd,  and  L  +,  when  x 
is  made  -  oo  the  value  of  the  first  memb'T  is  -  ;  and  when  x  is  0.  tlie  value  is  -t .  Hence  while 
X  passes  from  -  oo  to  0,  the  function  changes  sign,  and  hence  must  pass  through  0 :  i.  e.,  for 
Bome  value  of  x  between  -  oo  and  0,  the  equation  is  satisfied.  In  like  manner,  if  L  is  -,  when 
sc  -  0,  the  function  is  -,  and  when  ar  =  ^  oo  the  function  is  +  .  Hence  some  value  of  x  between 
-0  and  +  (X,  satisfies  the  equation.  It  follows  from  this  that  in  an  equation  of  an  odd  degree, 
If  the  absolute  term  is  +,  there  is  at  least  one  real,  negative  root ;  and  if  the  absolute  term  is  -, 
there  is  at  least  one  real,  jwsidve  root. 

If  n  is  even  and  Z,  -,  cc  =  0  makes  the  function  -  ,  and  x=  ±ra  makes  it  +.  Hence  while  x 
passes  from  -  oo  toO,  the  function  changes  sign  from  4-  to  -,  and  there  is  at  least  one  real, 
negative  root;  also,  while  x  passes  from  0  to  +  00,  the  function  changes  sign  from  -  to  +  ,  and 
there  is  at  least  one  real,  positive  root.  Therefore  every  equation  of  an  even  degree  in  which 
the  absolute  term  is  -,  has  at  least  two  real  roots,  one  negative,  and  one  positive. 

The  difficulty  occurs  in  proving  that  an  equation  of  an  even  degree  has  a  root  when  Z,  is  + . 
The  roots  of  such  an  equation  may  be  all  imaginary. 

t  This  i?  read  "  the  nth  cp  function  of  x." 


208  ADVANCED  COURSE  IN  ALGEBRA. 

Now,  as  x=  a,  or  x  =  b,  or  .r  —  any  one  of  tlu'  qmintitu*s  a,b,  e  -  -  -  -  I, 
will  render /(j)  equal  to  0,  each  one  of  the.se  will  ^utit^fy  the  equation  /(.r)=0. 
Therefore  this  equation  has  n  roots. 

Again,  since  it  is  evident  that  we  have  resolved  /(.r)  into  its  j^rimc  factors 
with  respect  to  x,  there  can  be  no  other  factor  of  the  form  x  —  m  in  J\.v),  hence 
no  other  root  of  /(t)=0,  and  this  whether  m  is  equal  to  one  or  more  of  the  roots 
a,b,c  -   ■  -  -  n,  or  not.     Therefore  /(.r)  =  0  has  only  n  roots. 

233.  Cor.  1. —  T/ie j^oli/nomial  x°+ Ax"-'+ 13x"-'+ Cx""' 

L,  or  f  (x),  =  (x  —  a)  (x  —  b)  (x  -  c)  -  -  -  -  (x  —  1),  in  which 
a,  b,  c  -  -  -  -    1  (fre  the  I'oofs  of  f  (x)  =  0. 

23H,  Cor.  :2. — The  equation  f  (x)  =  0  may  have  2,  3,  or  even  ii 
equal  roots,  as  there  is  no  inconsistence/  in  supposing  a  =  b,  iv  =  b 
=  c,  or  a  =  b  =  c  =    -  -  -  -     \^in  the  above  demonstration. 

237*  Cor.  3. — Imaginary  roots  enter  into  equations  having  only 
real  coefficients,  in  conjugate  pairs  (22a  ^  Part  I.)y  that  is,  (/'f(x)=0 
has  only  real  coefficients,  if  it  has  one  root  of  the  form  a  +  /JV—  1, 
it  has  another  of  the  form  a  —  ^V  —  1 ;  or,  if  it  has  one  of  the  form 
/^V—  1,  it  has  another  of  the  form  — /SV —  1- 

Tliia  ia  evident,  since  only  thus  can /(ar)=(ar— a)  (a:— &)  (.T—c)  -  -  -  -  {x—n); 
that  is,  if  one  root,  a  for  example,  is  a—/iV—\,  there  must  be  another  of  the 

form  a+fi  V—\,  in  order  that  the  product  of  these  two  factors  shall  not  involve 
an  imaginary.  Thus,  [T-{ix+ft  V^)]  x  [x-{a-/S  \^-l)]=x*-2ax+{a^+  ft'), 
a  real  quantity.  So  also  (x  —  ft  V'— 1)  {x  +ft  V—l)  =  x*+  ft*,  a  real  quantity. 
But  if  the  assumed  imaginary  roots  be  not  in  conjugate  pairs,  the  product  of  the 
factors  (x  —  a)  (.c —&)  (i^  —  c)  •  -  -  -  {x  —  /)  will  involve  imaginaries. 

238,  Cor.  4. — Hence  an  equation  of  an  odd  degree  must  have  at 
least  one  real  root  ^  but  an  equation  of  an  even  degree  does  not  9ieces- 
sarily  have  any  real  root. 

230*  Cor.  5. — If  an  equation  has  a  pair  of  imaginary  roots,  the 
known  quantities  entering  into  the  equation  mag  be  so  varied  that  the 
two  imaginary  roots  shall  first  give  place  to  two  equal  roots,  and  then 
these  to  two  real  and  unequal  roots 

As  shown  above,  imaginary  roots  arise  from  real  quadratic  factors  in  /(.p). 
l^t  x^—'Zax  +  &  be  such  a  quadratic  factor,  whence  .t*—  2ax  +  6  =:  0  satisfies 
fix)  =  0,  and  a  ±  Va*  —  b  are  the  corresponding  roots  of  f{x)  =  0.  Now,  if 
h  >  n-,  these  roots  are  imaginary.  If,  however,  b  diminishes  or  a  increases  (or 
lH)th  change  thus  together),  when  h  =  a^  the  two  imaginary  roots  disa])pear  and 
we  have  in  their  place  two  real  roots,  each  a.     If  the  same  change  in  a  and  b 


NUMERICAL   HIGHEU   EQUATIONS.  203 

continues,  so  that  a^  becomes  greater  than  h,  the  two  real,  equal  roots  in  turn 
give  place  to  two  real,  unequal  roots.  Now  as  a  and  b  are  functions  of  the  known 
quantities  of  the  equation  f{.i)  =  0,  such  changes  are  evidently  possible. 

240,  Sen.  1. — That  an  equation  has  a  number  of  roots  equal  to  its 
degree,  is  illustrated  geometrically  by  the  fact,  that,  if  we  write  y  =f{x)  and 
construct  the  locus,  we  shall  always  find  that  a  straight  line  can  be  drawn 
so  as  to  cut  the  locus  in  1  point  and  only  1,  if  /(a.)  is  of  the  1st  degree 
(Ex's.  1-13,  CiiAr.  II.);  in  2  and  only  2  points,  \i  f{x)  is  of  the  2d  degree 
(Ex's.  15-18,  Chap.  II.);  in  3  and  only  o  points,  if  f{x)  is  of  the  3d  degree 
(Ex's.  19-23,  Chap.  II.);  in  4  and  only  4  points,  if  /(.<)  is  of  the  4th  degree 
(Ex's.  24-28,  CuAP.  II.),  and  specially  illustrated  by  the  line  X/  X„  {Fig.  20), 
etc. 

241,  Scir.  2. — The  fact  that  imaginary  roots  enter  real  equations  in 
pairs  is  also  beautifully  illustrated  by  the  loci  of  equations.  Thus  the  equa- 
tion ^^— 3a; +  5=0  has  two  imaginary  roots,  and  no  real  roots.  Now,  by  ref- 
erence to  Fig.  9  of  the  preceding  chapter,  we  see  that  the  locus  of  y='x'^—'dx 
+  5  does  not  cut  the  axis  of  abscissas  at  all ;  i.  «.,  that  no  real  value  of  x  will 
giveyi;a-)=0.  But,  if  the  equation  were  so  modified  as  to  make  each  ordinate 
only  "V"  less  than  it  now  is,  /.  ^.,  if  y=x'^—Zx+"i^  we  should 
have  the  same  locus,  but  changed  in  position  so  as  just  to 
touch  the  axis  of  a*,  as  in  c,  thus  giving  f{x)—0  two  real  and 
equal  roots.  If,  again,  we  wrote  y—x:^—^x—"d,  we  should  have 
the  locus  referred  to  the  axis  A"X",  and  /(ic)  =  0  would  have 
two  real  and  unequal  roots.  Thus  we  see,  conversely,  how 
two  real,  unequal  roots  can  pass  into  two  real  and  equal  roots 
by  a  proper  change  in  the  equation,  and  how  by  a  further 
change  ttco  equal  real  roots  dlHajypear  at  a  time,  passing  into  two 
imaginary  roots  as  the  equation  changes  form.  All  that  is 
necessary  in  this  change  in  the  form  of  the  equation  is  a  pro- 
jier  change  in  the  absolute  term. 

Fig   32 
Again,   consider  Fig.  14,  and  the  corresponding  equation 

y=a;''— 6.r*+ll.T— G.     First  we  observe  that  as  this  locus  cuts  the  axis  of  x 

three  times,  there  are  three  real  roots.     Now  cliange  the  absolute  term  —6 

by  allowing  it  to  increase  gradually,  becoming  —51,  — 5^,  —5,  etc.     We  shall 

find  that  the  axis  of  x  moves  down,  and  the  two  roots  A  d  and  A/  approach 

equality,  first  becoming  equal  Avhen  the  axis  just  touches  the  lowest  point  e 

of  the  curve,  and  tlien  hoth  hecoming  imaginary  together. 

Or,  in  conclusion,  this  matter  is  illustrated  by  the  fact  that  whatever  the 
degree  of  the  equation /(.r)=0,  if  we  construct  the  locus  of  y=f{x),  we  shall 
find  that  we  can  draw  a  straight  line  which  will  cut  the  curve  in  a  number 
of  points  equal  to  the  degree  of  the  equation,  and  that  if  tlie  line  gradually 
moves  from  this  positi(m  so  as  to  cut  the  curve  in  any  less  number  of  points, 
it  will  always  be  found  first  to  run  two  intersections  together,  corresponding 


Y 

•\ 

J 

A 

X 

A 

X 

Y' 

210  ADVANCED  COURSE  IN  ALGEBRA. 

to  a  change  of  two  unequal  roots  into  two  equal  roots,  and  then  drop  out 
hoth  these  intersections,  corresponding  to  the  introduction  of  two  imaginary- 
roots  at  a  time. 


24:2,  I^rop. — If  the  equation  f(x)=:0  has  equal  roots,  the  highest 
common  divisor  of  f(x)  and  its  differential  coefficient*  f (x),  being 
2nit  equal  to  0,  co?istitutes  an  equation  which  has  for  its  roots  these 
equal  roots,  and  ?io  othe^  roots.\ 

Dem. — Let  a  be  one  of  the  m  equal  roots  of  f{x)=0,  and  let  the  other  roots  be 

b,c I;  then/(a-)=(a'-rt)'" ix-b){x^c) .  •  .  .  {x-  l){235).    Differentiating 

(152)  and  dividing  by  dx,  we  have 

f\x)=m{x-aY'-^{x-b){x-c)  -  -  -  -  {x-l)  +  {x-n)"'  (x-c) (x- I)  +  -  • 

.  .  .  -  +  (x—a)"  (x—b)  (x—c)  -  -  . .  +  etc. 
Now  (a?— a)*-'  is  evidently  the  highest  common  divisor  of  f{x)  and  f\x),  and 
{x—a)r~^  =0  is  an  equation  having  a  for  its  root,  and  having  no  other. 

In  a  similar  manner,  if  f{x)=0  has  two  sets  of  equal  roots,  so  that 

f{x)=(r-a)'"{x-hyix~c){x-d) (x-l), 

diflFerentiating  and  dividing  by  dx,  we  have 

f'{x)=m{T-a)'--^x-by{x~c){x-d) {x-l)+{x-a)'^r{x-^by-\x-c)  (x-d) 

(x-l) 

+(x-a)'^  (x--by  (x-d)  -  '  '  -  («— n)+(a;-rt)*(ar-6)''(«-c)  -  -  -  -  {x-l)+-- 

• ^{x—ay"{x—by{x—c){x—d)-  -  -  -  4-  etc. 

Now  the  highest  common  divisor  of /(x)  and /'(a?)  is  evidently  (x—a)"'-^{x—by-^. 
Putting  this  equal  to  0,  we  have  {x—a)''-^{x—by-^=0,  an  equation  which  is  sat- 
isfied by  x=a  and  x=b,  and  by  no  other  values. 

Thus  we  may  proceed  in  the  case  of  any  number  of  sets  of  equal  roots. 

243,  ScH. — In  searching  for  the  equal  roots  of  equations  of  high  degree, 
it  may  be  convenient  to  apply  the  process  of  the  proposition  several  times. 
Thus,  suppose  that  /(.r)=0  has  m  roots  each  equal  to  a,  and  r  roots  each 
equal  to  b.  Then  the  highest  common  divisor  of /(r)  and  f\x)  is  of  the  form 
{x—a)'*~^{;t^by~^\  whence  (jr_a)'"~'(j?— &)'-'=0  is  an  equation  having  the 
equal  roots  sought.  Therefore  we  can  find  the  highest  common  divisor  of 
{x—aT'^^x—bY'^  and  its  differential  coefficient  which  will  be  of  the  form 
{x—a)''~^{x—by~^i  and  write  {x—nY'~^(x—by-^z=0,  as  an  equation  containing 
the  roots  sought.  This  process  continued  will  cause  one  of  the  factors  (x—a) 
or  (x—b)  to  disappear  and  leave  ix—a)'*~'=Q,  when  m>r;  {x—by~"'—0, 
when  r>  m  ;  or  {x—a)(x—b)=0,  when  m—r.  From  any  one  of  these  forms 
we  can  readily  determine  a  root. 

*  The  differential  coefficient  of  a  fnnrtion  is  sometimes  called  its  first  derived  polynomial. 

t  The  student  most  not  suppose  that  the  roots  o{J\x)=0,  and  its  first  differential  coefficient 
/'(j')=0,  are  necessarily  alike.  f'(x)=&  series  of  temu  some  of  which  may  be  +  and  some  - ,  and 
which  may  destroy  each  other,  so  as  to  render/'(T)=0,  Ibr  other  values  of  x  than  such  as  render 
/{z)=0,  and  not  necessarily  for  any  which  do  render /(aj)=0,  except  the  equal  roots  of  the  latter. 


NUMERICAL     HIGHER     EQUATION?.  211 

,  244,  JProp. — In  an  equation  f(x)=:0,  f(x)  icill  change  sigyi  when 
X  passes  through  any  real  root,  if  there  is  but  one  such  root,  or  if  there 
is  an  odd  number  of  such  roots  ;  but  if  there  is  an  even  number  of 
such  roots,  f(x)  will  not  change  sign. 

Let  a,  b,  r  .  •  .  .  e   hv  the  roots  of  f{T)=0,  so  that  f{.i)={.v—n)  (x—b)  (x—c) 

(x—e)=0  {23t>).     Conceive  .i-  to  start  with  some  value  less  than  the  least 

root,  and  continuously  increase  till  it  becomes  greater  than  the  greatest  root. 
As  long  as  .r  is  less  than  the  least  root,  all  the  factors  .v—a,  x—b,  etc.,  are  nega- 
tive ;  but  w^hen  <ic  passes  the  value  of  the  least  root,  the  sign  of  the  factor  con- 
taining that  root  will  become  +,*  and  if  there  is  no  other  equal  root,  this  factor 
will  be  the  only  one  which  will  change  sign.  Hence  the  product  of  the  factors 
will  change  sign.  But  if  there  is  an  even  number  of  roots,  each  equal  to  this, 
an  even  number  of  factors  will  change  sign  ;  whence  there  will  be  no  change  in 
the  sigu  af  the  function.  If,  however,  there  is  an  odd  number  of  equal  roots, 
tUe  passage  of  x  through  the  value  of  this  root  will  cause  a  change  of  sign  in  an 
odd  number  of  factors,  and  hence  will  change  the  sign  of  the  function. 

Finally,  as  it  ia  evident  that  the  signs  of  the  factors,  and  hence  of  the  function, 
will  remain  the  same  while  x  passes  from  one  root  to  another,  and  in  all  cases 
changes  or  does  not  change  as  above  when  x  passes  through  a  root,  tlie  j^roposi- 
tion  ia  established. 

245,  ScH. — This  pro])osition  is  illustrated  by  putting  y—f{:i)  and  con- 
structing the  locus,  as  in  the  preceding  chapter.  Thus,  Ex.  15,  Fig.  6.  In 
this  case  y=f{x)=x'-^rX—iS.  The  least  root  is  —3.  When  x  is  less  than  —3, 
as  —4,  or  — 3.^  {anything  less  than  —3),  y,  or/(.r),  is  +.  When  x  is  —3,  ?/, 
or/(.r)=0,  and  the  equation /(.r)=0  is  satisfied,  and  —3  is  a  root  of  the  equa- 
tion. When  X  becomes  greater  than  —3,  as  —2,  y,  or  /(^),  becomes  nega- 
tive, changing  sign  when  x  passes  through  the  value  of  tlie  root  —3.  As  x 
increases,  y,  or  /(.r),  remains  — ,  till  x  reaches  -f  2,  at  which  value  of  .r, 
y,-/(2-)=r:0,  and  the  equation /(.r)=0  is  satisfied.  When  x  passes  this  value, 
becoming  anytliing  greater  than  2,  y,  or /(.r),  becomes  +,  i.  c,  changes  sign 
as  x  passes  through  the  root  2.  The  same  thing  is  illustrated  by  the  loci 
in  Figs.  7,  11,  13,  14,  15,  and  18,  with  their  corresponding  equations. 

That/(.r)  does  not  change  sign  upon  .t's  passing  through  the  value  of  one 
of  two  equal  roots  of  /(.r)=0,  is  illustrated  in  Fig.  8  and  its  corresponding 
equation,  Ex.  17.  In  this  case  yz^f\x)-x^ -L%+^,  and  the  equation 
a;2_4^+4=0  has  two  roots  each  equal  to  2.  Now  when  x  is  anything  less 
than  2,  y,  i.  e.  f{x\  is  +  ;  when  x=2,  y,  or  f{x\  is  0,  and  the  equation  /(.r)=rO 
is  satisfied.  But  when  x  passes  the  value  2,  /(.t)  does  not  change  sign  ;  it 
remains  +.  The  same  truth  is  illustrated  by  the  loci  in  Figs.  10,  13,  16, 
and  17,  and  their  corresponding  equations.  Fig.  16  illustrates  the  case  in 
which  there  arc  two  pairs  of  equal  roots. 


♦  SnppoBC  c  be  the  Kast  root,  and  that  c'  is  the  next  state  of  x  greater  than  c;  then  c'- 


212  ADVANCED  COUKSE  IN  ALGEBRA. 

Ex.  29  will  be  found  very  instructive.  The  locus  in  Fig.  20  illustrates  the 
case  of  3  equal  roots.  Here  y  =/(.r)  =  r'*-f- 4a!*— 14a;-—  ITa;  —  6.  The  least 
root  is  —  3.  When  a;  <  —  3,  f{x)  is  —  ;  when x—  —  3,/(.r)  =  0 ;  when x passes 
—  3,  increasing,  /(.r)  changes  from  —  to  +,  and  remains  +  till  x  =  —  1,  when  it 
becomes  0,  and  changes  sign  as  x  passes—  1,  noUcithstanding  there  are  equal 
roots.    But  there  is  an  odd  number  of  such  roots,  viz.,  three. 

Thus,  if  X,'X,  were  to  revolve  about  ^;' until  it  took  the  position  X'X,  the 
intersections  b'  and  d'  would  run  into  c,  the  three  intersections  becoming  one. 


246.  Prop. —  Changing  the  signs  of  the  terms  of  an  equation 
containing  the  odd  powers  of  the  xinhnown  quantity  changes  the  signs 
of  the  roots, 

De.m. — If  x  =  a  satisfies  the  equation  x^—  Ax^-\-  Bx^—  Ca;  +  i>  =  0,  we  have 
a^—Aa*-\-  Ba^—  Cu-\- 1)  =  0.  Now  changing  the  signs  of  the  terms  containing 
the  odd  powers  of  x,  we  have  x^—  Ax*—  Bx^-\-  Cx  -\-  D  —  0.  This  is  satisfied 
by  a;  =  —  a,  if  the  former  is  by  x  =  a.  For,  substituting  —  a  for  x,  we  have 
a^—Aa*+Ba^—  Ca  +  D  =  0,  the  same  as  in  the  first  instance. 

247*  Cor. —  Changing  the  signs  of  the  terms  containing  the  eve?i 
powers  will  answer  equally  well,  since  it  amounts  to  the  same  thing; 
and  if  tee  are  careful  to  put  the  equation  in  the  complete  fortn, 
changing  the  signs  of  the  alternate  terms  xciU  accomplish  the  purpose, 

III. — The  negative  roots  of  a;^  —  7a;  4-  6  =  0,  are  the  positive  roots  of  —  a;' 
4-  7a;  +  6  =  0,  or  of  a;'—  7a;  —  6  =  0  (0  being  considered  an  even  exponent) ;  or, 
writing  the  equation  a;' ±  Ox*— 7a;  +  6  =0,  changing  the  signs  of  alternate 
terms,  and  then  dropping  the  term  with  its  coeflBcient  0,  we  obtain  the  same 
result. 

Again,  the  negative  roots  of  x*"—  Ix^—rtx^-h  8a;'—  132a;*  +  508a;  —  240  =  0, 
are  the  positive  roots  of  x'^+  Ix^—^x*—  Sx^—  132a;*—  508a;  —  240  =  0,  or  of 
-  x^-  7x'  +  5a;*+  8a;'+  132a;*+  508a;  +  240  =  0. 


248.  Prob. —  To  evaluate  *  f  (x)/or  any  particular  value  of  x, 
cw  X  =  a,  more  expeditiously  than  by  direct  substitution. 

Solution.— As  f{x)  is  of  the  form  a?»  +  ^a?»-'  +  5a;"-'  +  Ca;«-«  -  -  -  -  L, 
let  it  l>e  required  to  evaluate  x*  +  Ax^-if  Bx^+  Ox  4-  D  for  x  =  a.  Write  the 
detached  coefficients  as  below,  with  a  at  the  right  in  the  form  of  a  divisor  :  thus 

1     +^  +B  +C  +D  \a_ 

a  a*+Aa  a^+Aa^+Ba  a*+Aa^+Ba'^  +  Ca 


a+A         a*  +  Aa+B        a*-\-Aa* -{-Ba+C        a*+Aa^-]-Ba^  +  Ca+l) 

*  This  means  to  find  the  value  of.  Thus,  snppose  we  want  to  find  the  value  of  a;6-53?'' 
+  8iC*-8a?»+  6a;«-x-  18,  fora;  =  .5.  We  might  pubptitnte  5  forr,  of  conrpc,  and  accomplish  the 
end.    Bui  there  is  a  more  expeditious  way,  as  the  solution  of  ihis  probletn  ^ill  show. 


NUMERICAL    HIGHER   EQUATIONS.  213 

Having  written  the  detaclied  coefficients,  and  the  quantity  a  for  which  f{x)  is 
to  be  eviiluated  as  directed,  multiply  the  first  coefficient  1  by  a,  write  the 
result  under  the  second,  and  add,  giving  a  +  A.  Multiply  this  sum  by  a,  write 
the  product  under  the  third  coefficient  B,  and  add,  giving  a^  +  Aa  +  B.  In  like 
manner  continue  till  all  the  coefficients  (including  the  absolute  term,  which  is 
the  coefficient  of  x^)  have  been  used,  and  we  obtain  a^+  Aa-^+  Ba^+  Ca  +  D, 
which  is  the  value  of  f{x)  for  x  =  a. 

Illustration.— To  evaluate  x^—5x'^+  2x^—  3x^+  Qx^  —  x  —  12,  for  x  =  5t 


-5           +2 

-3 

+6            -1 

-12  1  5 

5              0 

10 

35           205 

1020 

0               2 

7 

41            204 

1008 

he  value  of  x^  — 

■5x'+2x*- 

.'dx'+Qx^-x- 

12,  for  a;  =.5;  and  it 

is?  easy  to  see  that  much  labor  is  saved  by  this  process. 

"We  arc  now  prepared  for  the  solution  of  the  following  important 
practical  problem : 

240,  I^vob, —  To  find  the  commeiimirahle  roots  of  numerical 
higher  equations. 

The  solution  of  this  problem  we  will  illustrate  by  practical  examples. 

Examples. 
1.  Find   the   commensurable  roots  of  re*— 2a;''— 15.r^+ 8a:'+  68a; 
+  48  =  0,  if  it  has  any. 

Solution. — By  (232),  if  this  equation  has  any  commensurable  roots  they 
are  integral : — it  can  have  no  fractional  roots. 

Again,  by  (230),  the  roots  of  this  equation  with  their  signs  changed  are  fac- 
tors of  48.  Now,  the  integral  factors  of  48  are  1,  2,  8,  4,  C,  8,  12,  16,  24,  48. 
Hence,  if  the  equation  has  commensurable  roots,  they  are  some  of  these  num- 
bers, with  either  the  -f  or  —  sign.  We  will,  therefore,  proceed  to  evaluate 
fix)  (t.  e.,  in  this  case  x'^-2x*-  15.r '-l-  Sx^+  (iSx  +  48),  ior  x=  +  l,x=  —  1, 
x=  -\-2,x=  —2,  etc.,  by  {248),  as  follows  : 

1     -2  -15  +  8  +68  +48  [  +1 

1  _  1  _16  -  8  60 


-1  -16  -  8  60  108 

Hence  we  see  that  for  x=  +  1,  f{x)  =  108,  and  +1  is  not  a  root  of  f{x)  =  0. 
Trying  x  =  —  1,  we  have 

1     -2  -15  +  8  +68  +48  |  -1_ 

-1  3  12  -20  -48 


-3  -12  20  48  0 

Thus  we  see  that  for  a;  =  —  l,f{x)  —  0,  and  hence  that  —  1  is  a  root  of  our  equ*. 
tion. 


214  ADVANCED  COURSE  IN  ALGEBRA. 

We  might  now  divide /(a*)  by  .r-t-1  {2:il)  and  reduce  the  degree  of  the  equa- 
tion  by  unity.  But  it  will  be  more  expeditious  to  proceed  with  our  trial.  Let 
us  therefore  evaluate/(.r)  for  .v=  +2.     Thus  : 

1     _2  -15  +8  +C8  +48  I  +2 

2  0  -30  -44  +48 


0  -15  -22  24  9G 

Hence  for  x=  +2,  f{x)=dQ,  and  +2  is  not  a  root.     Trying  x=—2,  we  have 

1     -2  -15  +  8  +08  +48  I  -2 

-2  8  14  -44  -48 

-4-7  22  24  0 

Hence  for  x-  —2,f{x)=0,  and  —2  is  a  root.     Trying  x=+S,  we  have 


1     _2            -15            +  8 

+08 

+48  1  +3 

8                3-30 

-84 

-48 

1            -12            -28 

-10 

0 

Hence  for  x  =  +o,  f{x)=0,  and  +3  is  a  root. 

Trying  x= 

—3,  we  have 

1     -2            -15            +  8 

+08 

+  48  1  -3 

-3               15                 0 

-24 

-132 

-5                 0                 8 

44 

-  84 

Hence  for  x=  -S,f{x)zz-S4,  and  -3  is  not  i 

!i  root.    Tryi 

ing  ar=4,  we  1 

1     _2            -15            +  8 

+08 

+  48  1    4» 

4                 8            -28 

-80 

-48 

2  -  7  -20  -12  0 

Hence  for  x=4,f{x)=0,  and  4  is  a  root. 

We  have  now  found  four  of  the  roots,  viz.,  —1,  —2,  3,  and  4.  Their  product 
with  their  signs  changed  is  24.  Hence,  by  {2:i0)  48-s-24=2  is  the  otlier  root 
with  its  sign  changed,  i.  e.  there  are  two  roots  —2. 

That  our  equation  had  equal  roots  could  have  been  ascertained  by  the  princi- 
ple in  {242) ;  but  as  the  process  of  finding  the  H.  C.  D.  is  tedious,  it  is  generally 
best  to  avoid  it  in  practice. 

2  to  12.  Find  the  roots  of  tlie  following : 

(2.)  0^  -  rc»  -  39.7-'  +  Ux  +  180  =  0 ; 

(3.)  x"  +  ox'  -  9x-  -  45  =  0 ; 

(4.)  a:*  +  2a:*  -  23.r  -  00  =  0 ; 

(5.)  a^  -  3.4-^  -  Ux'  4-  48.T  -  32  =  0 ; 

(6.)  x'  -Sx'  +  13.7:-  G  =  0; 


*  Of  course  it  it*  not  iieccHsary  to  relaui  the  -f-  sign,  as  we  have  done  in  the  preceding  opera- 
tions: it  has  been  done  simply  for  emphasis. 


NUMERICAL  HIGHEIi   EQUATIONS.  215 

(7.)  a;^  -  Ux'  +  18a;  -  8  =  0 ;  * 

(8.)  x^  -  Sx'  +  6.r^  -3x^  —  Sx  -\-  2  =  0; 

(9.)  x'  —  I'dx*  +  Glx""  -  171a;'  +  21Ga;  —  108  =  0 ; 

(10.)  x'  -  45ar^  -  40a;  +  84  =  0  ; 

(11.)  x^  —  3:c*  -  9a;^  +  'ilx'  -  10a;  +  24  =  0  ; 

(12.)  x'-W  +  11a;*  -  Tx'  +  14a;^  -  28a;  +  40  =  0. 

13  to  20.  Apply  the  process  for  finding  equal  roots  (242 ^  243)  to 
the  following : 

(13.)  x""  +  S.c'  +  20a;  +  16  =  0; 

(14.)  x"  -  a;'-  8a;  +  12  =  0  ; 

(15.)  x'  -  ox'  -  8a;  +  48  =  0  ; 

(16.)  u^  -  lla;^  +  18a;  -  8  =  0  •, 

(17.)  X*  +  13x^  +  3dx'  +  31a;  +  10  =  0 ; 

(18.)  x'  -  13x*  +  67a.'=*  -  171a;^  +  216a;  -  108  =  0; 

(19.)  x'  +  3ar^  -  6x*  -  Gx^  +  9a;^  +  3a;  -  4  =  0 ; 

(20.)  a;'+  5a;''4-  6ar^-  Qx'-  lox'-  3x'+  Sx  +  4  =  0.     (See  243.) 

21  to  27.  Having  found  all  but  two  of  the  roots  of  each  of  the  fol- 
lowing by  (248),  reduce  the  equation  to  a  quadratic  by  (231),  and 
from  this  quadratic  find  the  remaining  roots : 

(21.)  u^  -  Qx'  +  lOa;  -  8  =  0  ; 

(22.)  a.-*  -  4a;^  -  8a;  +  32  =  0 ; 

(23.)  ar'  -  3.a;2  +  a;  +  2  =  0 ; 

(24.)  x'  -  (jx^  +  24a;  -16  =  0; 

(25.)  x'  -  12ar'  +  50:c^  -  84a;  +  49  =  0  ;  f 

(26.)  a-*  -  9ar»  +  17a;^  +  27a;  -  60  =  0  ; 

(27.)  x'  -  4a;*  -  16:6-^  +  112ar'  -  208a;  +  128  =  0. 


28  to  34.  Apply  the  processes  of  (228)  to  reduce  the  following  io 

the  form  x*  +  Jia;""*  +  Bx""^  +  (7a;"~^ L  =  0,  before  searchmg 

for  roots : 

{^^^j  2ar^  -  3a;'  +  2a;  -  3  =  0  ;  t 

*  In  order  to  apply  the  process  of  evaluation,  the  coofflcieuts  of  the  missing  powers  must  be 
Bupplicd.    Thus  we  have  1  +0  -  11  +  18  -S. 

t  Apply  the  method  for  fliiding  equal  roots.  The  method  of  trial  based  upon  {230)  as  applied 
by  {248)  is  likely  to  lead  to  much  unnecessary  work  when  there  are  several  equal  roots,  and  all 
the  others  incommensurable. 

$  Wehavea:»-|ar2+ar-^=0.     Putar=|,   whence  l^-gj^y*  +  ^y-g=0,orya_  — y2 +^2y 

-  —  =  0.  If  now  i=2,  we  have  y3  -3y«-f  4y-12=0,  which  can  be  solved  as  before,  for  one  value, 
of  y.and  the  equation  then  reduced  to  a  quadratic  and  solved  for  the  other  values.  Finally, 
remembering  that  a:= , J  y,  we  have  the  values  of  a;  required. 


216 


ADVANCED  COUKSE  IN  ALGEBUA. 


(29.)  3a:*  — 2ar'  _  6a;  +  4  =  0 ; 

(30.)  8^  -  2Gx'  +  11a;  +  10  =  0; 

(31.)  .r*  —  ia;  +  3^  =  0 ;     (Look  out  for  equal  roots.) 

(32.)  x'-63^+  OJa;''  -  3a:  +  4i  =  0 ; 

(33.)  X 


19a; 


(34.) 


/ 


■y-?- 


403a;- 


a;'  -  3a;  +  22 


~i  =  n-  +  ^) 


250.  By  means  of  the  ])roperty  exhibited  in  (235)  produce  the 
equations  whose  roots  are  given  in  the  following  ex^jimples  : 


1.  Roots  1,  -3,  4. 

2.  Roots  ^2,  —a/2,  —1,  3. 

3.  Roots  1,  2,  2,  -3,  4. 

4.  Roots  -3,  2  +  \/^,  2-a/^1. 

5.  Roots  3,  —2,  —2,  —2,  1. 

6.  Roots  |,  J,  -|. 

7.  Roots  l=b\/i:2,  2=fcA/^. 


8.  Roots  li,  2,  ^3,  -V3. 

9.  Roots  a/— ^,     —  a/^,     a/5, 

10.  Roots  10,  -13,  i,  1. 

11.  Roots      3-2a/3,        3  +  2a/3", 

2-3a/^,    2  +  3a/^,    1, 
-1. 


SECTION  IL 


SOLUTION   OF    NUMERICAL    HIGHER    EQUATIONS    HAVING    REAL, 
INCOMMENSURABLE,    OR    IRRATIONAL    ROOTS. 


251.  As  all  equations  having  real  roots  have  real  coefficients* 
(237),  and  as  all  such  can  be  reduced  to  the  form  a;"  +  ^af"* 
+  i5af "'  +  Ca;""'  -  -  -  -  2/  =  0,  which  we  represent  by  f(x)  =  0 
(229),  we  shall  consider  this  as  the  typical  form.  Moreover,  since, 
if  an  equation  of  this  character  has  equal  roots,  they  can  be  deter- 
mined by  (242,  243),  and  the  degree  of  the  equation  depressed 
by  (231),  we  need  only  to  consider  the  case  in  which /(a*)  =  0  has 
no  equal  roots. 


*  This  Is  evident  from  the  fact  that  f(x)={x-a)  (x-b)  (x—c)  -  -  -  -  (a*-n)=0.  In  which  if 
a,  6,  c,  -  •  -  -  n  are  real,  no  ima]^nary  quantity  will  be  fonnd  in  the  product  of  the  binomials. 


217 

2S2,  The  best  general  method  of  approximating  the  real,  incom- 
mensurable roots  of  such  equations,  is : 

1st.  To  lind  the  number  and  situation  of  such  roots  by  Sturm's 
Theorem  and  the  method  based  on  it. 

2d.  Having  found  the  first  figure  or  figures  of  such  a  root  by 
Sturm's  method,  to  carry  forward  the  approximation  to  any  re- 
quired degree  of  accuracy  by  Horner's  method  of  approxima- 
tion. 

These  methods  we  will  now  proceed  to  develop. 


Sturm's  Theorem  and  Method. 

253,  Sturm's  Theorem  is  a  theorem  by  means  of  which  we 
are  enabled  to  find  the  7iumher  and  sitiiatwn  of  the  real  roots  of  any 
numerical  equation  with  a  single  unknown  quantity,  real  and 
rational  coefficients,  and  without  equal  roots.* 

III. — Thus,  if  we  have  the  equation  a;^— 7a; +  7  =  0,  Sturm's  Theorem 
enables  us  to  determine  that  it  has  three  real  roots,  i.  e.,  that  all  its  roots  are 
real.  It  also  enables  us  to  ascertain  that  one  root  lies  between  1.3  and  1.4, 
anotlier  between  1.6  and  1.7,  and  the  third  between  —3  and  —4.  Hence  it  shows 
us  that  the  roots  are  1.3+ ,  1.G+,  and  —3.  with  a  decimal  fraction. 

254,  ScH. — Of  course  it  follows  from  the  above  that  if  the  equation  has 
commensurable  (227)  roots,  Sturm's  Theorem  will  enable  us  to  find  them, 
or  even  when  the  roots  are  not  commensurable,  it  will  enable  us  to  find  any 
number  of  initial  figures.  Thus  in  the  equation  x^  —  Ix  +  7  =  0,  we  might 
by  Sturm's  Theorem  find  that  the  first  root  is  1.35689+ ;  but  it  would  be 
too  tedious  an  operation  to  be  of  any  practical  utility,  as  will  appear  hereaf- 
ter. Wc  use  this  theorem  only  to  find  one  or  two  of  the  initial  figures,  or, 
enough  of  the  figures  to  enable  us  to  distinguish  between  (separate)  the  roots. 
Thus,  if  wc  had  an  equation /(a)  =  0,  of  which  two  roots  were  2.356873+  and 
2.3569564,  we  might  use  Sturm's  Theorem  to  find  the  first  five  figures  of  each 
root,  i.  €.,  to  distinguish  between  (separate)  the  roots;  but  this  is  not  the 
l>e8t  practical  method,  as  will  appear  hereafter, 

255,  The  Sturmian  Functions  of  /(.r)  =  0  (v/hich  has 
no  equal  roots)  are  functions  obtained  by  treating  f{x)  and  its  first 
differential  coefficient  f'{x),  as  in  the  process  of  finding  their  H.  C. 
D.,  except  that  -in  the  process  we  must  not  multiply  or  divide  by  a 
negative  quantity,  and  the  signs  of  the  several  remainders  must  be 

*  If  the  equation  which  we  wish  to  solve  has  equal  roots,  Ihoy  can  be  discpverpfl  by 
{242,  34itj,  and  the  degree  of  the  cqiiatiou  reduced  by  (liyi.>^ipji. 


HM  advanced  coukse  in  algebra. 

changed  before  they  are  used  as  divisors.     These  remainders  with 
their  signs  changed  are  the  Sturmian  Functions.''^ 

III.— Let  the  equation  f{x)  =  0  be  *^  —  4x*  —  a;  +  4  =  0.  The  first  differential 
coefficient  of  ;c^  —  4.«*  —  .c  +  4  is  3x*  —  8^-  —  1.  Dividing  x^  —  Ax'^  —  .r  -f  4  by 
3.g«_  8i^  _  ]^  first  multiplying  the  former  by  3  to  avoid  fractions,!  exactly  as  in 
the  process  of  finding  the  H.  C  D.,  we  find  the  first  remainder  of  lower  degree 
tlian  our  divisor  to  be  —  XSix  +  16,  Hence  19jj  —  16  is  the  first  Sturmian  Fhinc- 
tion  of  x^  —  4j:*  —  x-\-A.  Again,  dividing  '6x^—  8aj  +  1  by  19a;  —  16  (introducing 
puch  constant  factors  as  necessary),  we  find  the  next  remainder  to  be  —  2025. 
Hence  2025  is  the  second  Sturmian  Function  of  x^—  Ax*  —  x  +  4. 

2oG,  Wofation, — As  the  function  which  constitutes  the  first 
member  of  our  equation  is  represented  by  f(x),  and  its  first  differ- 
ential coefficient  by  f'{x),  we  shall  represent  the  Sturmian  Func- 
tio7is  by  fi(x)f  /i(:i')>  /3(^)>  ^tc,  read  "/sub  1  function  of  x"  "/sub 
2  function  of  ar,"  etc.,  or  simply  "  function  sub  1,"  "  function  sub  2," 
etc. 


2S7>  In  any  series  of  quantities  distinguished  as  +  and  — ,  a 
succession  of  two  like  signs  is  called  a  Permanence  of  signs,  and  a 
succession  of  two  unlike  signs  a  Variation. 

III. — In  the  function  x^  —  3j'''  —  2j-*  +  a''  +  x*  -\-  fix  —  A,  the  signs  of  the  terms 
are 

+     --     +     +     +     -. 

The  first  and  second  constitute  a  variation ;  the  second  and  third  a  perma- 
nence ;  the  third  and  fourth  a  variation  ;  the  fourth  and  fifth  a  permanence ;  the 
fifth  and  sixth  a  permanence  ;  and  the  sixth  and  seventh  a  variation.  Thus,  in 
this  case,  there  are  three  permanences  and  three  variations  of  signs. 

So  also  if  we  have 

f{x)  =  a;'  -  Ix*'  +  13ar3+  ^*  _  i^jj  +  4, 

fix)  =  5a;*  -  28a;-»+  39a;«+  2a;  -  16, 

/,(ar)  =  1  l.c^  -  4ac«  +  51a;  +  2, 

/,(a;)  =  3a;='-8a:  +  4, 

/3(a-)  =  a;-2, 

A{x)=0. 
For  X  =  0,  fix)  =  +  4,    or  f{x)  is  +  ;  f\x)  is  -  ;  /.(a-)  is  +  ;  /,(a;)  is  +  ; 
fi{x)  is   —  ;   and  f^{x)  being  0,  its  sign  is  not  considered.     Hence  the  series  of 
signs  of  these  functions,  for  a;  =  0,  is  +   —    +   +   —  ;  and  has  three  variations 
and  one  permanence. 

*  I  have  thought  it  best  not  to  include  /(x)  and  f'{x)  under  the  term-  Sturmian  Functions. 
There  seems  to  be  no  propriety  in  iiidudini,'  them,  ina:*much  as  they  are  not  peculiar  to 
Stnnn's  method  ;  and  by  excluding  them  an  important  distinction  is  marked. 

t  We  introduce  or  reject  constant  factors,  just  as  in  finding  the  H.  C.  D.,  only  vvc  may  not 
introduce  or  reject  n^^o^tu^  factors,  since  the  su/ns  are  an  essential  thinj;  in  these  functions,  and 
to  multiply  or  divide  by  a  negative  number  would  change  the  signs  of  the  functions. 


STURM  S  THEOREM  AND  METHOD.  219 

For  ar  =  1,  we  find  f{x),  -  ;  f\x),  +  ;  f,{x),  +  ;  f,{x),  -  ;  and  f^{x),  -  ;  the 
series  of  signs  being  -  +  h .  This  gives  two  variations  and  two  per- 
manences. 


258.  Prop.—Li  the  series  of  functions  f(x),  f'(x),  f,(x),  f,{x), 

f3(x),  f4(x),  f5(x) f„(x),  when  f  (x)  —  0  has  no  equal  roots,  if 

X  be  conceived  to  pass  through  all  possible  real  values,  that  is,  to  vary 
continuously,  from  —  oo  to  +qo  ,  there  loill  be  no  change  in  the  number 
of  variations  and  permanences  in  the  signs  of  the  functions,  except 
when  X  jxJ^ses  through  a  root  of  f{x)  =0;  and  when  it  does  pass 
through  such  a  root,  there  loiU  be  a  loss  of  one  variation,  and  only 
ojie,* 

Dem. — 1st.  Any  change  in  x  which  does  not  cause  some  one  of  the  functions 
to  vanish,  cannot  cause  any  change  in  the  signs  of  the  functions  ;  for  no  function 
can  change  its  sign  without  passing  through  0  or  oo ,  and  from  the  form  of  the 
functions  which  we  are  considering,  they  cannot  be  go  for  any  finite  value  of  x. 
(These  functions  are  all  of  the  form  Ax''  +  Hx"-'^  +  Cx~'^ L.) 

2d.  No  tijoo  consemitive  functions  can  vanish,  i.  e.,  become  0,  for  the  same  value 
of  J..  For,  in  the  process  of  producing  the  Sturmian  functions  f rom /(.t)  and 
f\x),  let  the  several  quotients  be  represented  by  q,  q',  q",  q"\  q^^,  etc. ;  whence, 
by  the  principles  of  division,  we  have 

f{^)=f'{^)q    -Mx),  (1) 

f'{x)^f,{x)q'  -f,{a^  (2) 

A(^)=A{-i')q"  ~f.(^),  (3) 

A{^)=fA^yr-A{x),  (4) 

Ux)=f\{x)q^--f,{x),  (5) 
etc.,         etc.,         etc. 

Now,  if  possible,  suppose  that  some  value  of  x,  sls  x  =  a,  renders  two  consecutive 
functions,  as  fii-i-)  and  fi{x)  each  0 ;  that  is,  that  they  vanish  simultaneously. 
Then,  since  from  (4)  we  have  f^ix)  =fAx)q"'  —  fi{x),  fi{x)  =  0.  So,  also,  from 
(5),  fsix)  =f*{x)q'''  —  fi{x),  and  f^ix)  =  0.  Thus, as  a  consequence  of  the  simul- 
taneous vanishing  of  any  two  consecutive  functions,  we  could  show  that  all  the 
functions  would  vanish.  But  as,  by  hypothesis,  f{x)  and  /'(•'')  have  no  common 
divisor  containing  x,  the  last  remainder  found  by  the  process  of  finding  the 
H.  C.  D.  cannot  contain  x,  and  hence  cannot  vanish  for  any  value  of  x.  It  is 
therefore  impossible  that  any  two  consecutive  functions  of  the  series  should 
vanish  for  the  same  value  of  x  {i.  e.,  simultaneously). 

3d.   }V7i€n  any  one  of  the  functions,  except  f  (x),  vanishes  for  a  particular  value 

*  This  is  the  sabstancc,  though  not  the  exact  form,  of  the  celebrated  theorem  discovered  by 
M.  Sturm  in  1829,  and  for  which  he  received  the  mathematical  prize  of  the  French  Academy  of 
Sciences  in  1S34.  It  is  certainly  one  of  the  most  elegant  discoveries  in  algebraic  anal}>is  made 
iu  modern  times.  It  is  a  masterpiece  of  logic,  and  a  monument  to  tlie  sagacity  of  its  di>coverer. 
The  original  memoir  containing  this  theorem  is  found  in  the  "Memoires  prescntes  par  divert 
Bftvants  a  1' Academic  des  Sciences,"  Tom.  VI.,  1835. 


220  ADVANCED  COUKSE  IN  ALGEBRA. 

of  X,  the  adjacent  functions  have  opposite  signs  for  this  value.  Thus,  if  f^ix)  is  0 
for  X  =  b,  we  have,  from  {i),fs{i)  =  — /4(-f).  *•  «•»  the  adjacent  functions,  neither 
of  which  can  vanisli  for  tliis  value  (2d),  liave  opposite  signs. 

4th.  When  any  value  of  x,  as  x  =  c,  causes  any  function  except  f  (x)  to  vanish, 
the  number  of  variations  and  peiinanences  of  the  signs  of  the  functions  is  the  satne 
for  the  jtreceding  and  the  succeeding  values  ofs.,!.  e.,for  x  =  c  —  h  and  x  =  c  +  h, 
h  being  an  infinitesimal.  Tlius,  let  x  =  c  render  fi{^)  —  0  ;  then,  since  the  adja- 
cent  functions  liave  opposite  signs  for  this  value  of  x,  we  have  either  +fi{x),  0, 
— /4W.  or  —fi{x),  0,  +f4{x),  i.e.,  +,  0,  — ,  or  — ,  0,  -t-  (3d).  Again,  as  neither 
of  tliese  adjacent  functions  vanishes  for  x  =  c  (3d),  neither  of  them  can  change 
eign  as  x  passes  through  e  (1st).  But  fzix)  may  or  may  not  change  sign  as  x 
passes  through  c  {244) ;  hence  its  signs  may  be  4^,  =,  ± ,  or  T ,  the  upper  sign 
representing  the  sign  of /3(.e)  just  before  x  reaches  c,  and  the  lower  its  sign  just 
after  it  passes,  i.  c.,  for  x  =  c  —  h,  and  x=z  e+  h,  re8i>ectively.  Hence  all  tlio 
clianges  in  signs  which  can  occur  are  represented  thus:  +  ^  —,  -\-  =  —, 
+  ±  — ,  4-  T  — ,  —  ^  +,  —  =  +,  —  ±  +,  and  —  T  +.  These  taken  in 
any  way  give  simply  one  permanence  and  one  variation.  Jlenee  there  can  he  no 
change  in  the  number  of  variations  and  permanences  of  the  signs  of  t?ie  functions, 
consequent  upon  the  vanishing  of  any  intermediate  function. 

5th.  We  are  ixott  to  examine  what  changes,  if  any,  arc;  produced  in  the  num- 
ber of  variations  and  permanences  by  tlie  vanishing  of  an  extreme  function. 
And  in  the  first  place  we  repeat  that  the  last  function  cannot  vanish  for  any 
value  of  X,  as  it  does  not  contain  x.  We  have  then  to  examine  only  the  case  in 
which /(r)  vanishes,  i.  e.,  when  x  passes  through  any  root  of  f{x)  =  0.  For  this 
purpose  let  us  develop  f(x  +  h)  by  Taylor's  Formula,  considering  7i  an  infinitesi- 
mal.   Thus, 

fix  +  h)  =f{x)  +fXc)h  +/"(^) Y  4- /'"(.r) ^^  4-  etc. 

Now  let  r  be  any  root  of  f{x)  =  0,  and  substitute  in  this  development  r  for  x ; 
whence 

/(r  +  h)  =/(r)  +r{r)h  ^f'\r)  ^  +  f"\r)  ^  +  etc. 

As  r  is  a  root  of  f{x)  =  0,  /(?•)  =  0 ;  and  as  h  is  an  infinitesimal,  the  terms  con- 
taining its  higher  powers  may  be  dropped  {lUlf  and  foot-note).  Thur  we  have 
/(r  4-  A)  =  f'{r)h.  Hence,  as  A  is  4-,  we  see  that  f{r  4-  h),  that  is  the  function 
just  after  x  passes  a  root,  has  the  same  sign  as  f'{r),  i.  e.  f'(x)  when  a;  is  at  a 
root.  But  as  /'(r)  does  not  vanish  when  x  =  r  (3d),  f'{r  —  h),  f'{r),  and 
fir  4-  A)  have  the  same  signs.*  Again,  since,  by  hypothesis,  f{x)  =  0  has  no 
equal  roots,  it  changes  sign  wlien  x  passes  through  a  root  {244),  i.  e.,  f{r  —  h) 
and  f{r  4-  /<)  have  different  signs.  Thus,  as  f{x)  and  f'{v)  have  like  signs  just 
after  x  has  passed  a  root,  and  /(/)  changes  sign  in  passing,  while  f'{x)  does  not, 
these  functions  have  unlike  signs  just  l^efore  x  reaches  a  root,f  and  what  was  a 
variation  in  signs  becomes  a  permanence  ;  that  is,  a  variation  is  lost. 


*  That  18,  the  first  differential  coefficient  of  /(r)  docs  not  change  sign  when  x  pasecs  through 
arootof /(x)=0. 

t  From  this  we  see  that  the  roots  of  /'(x)=0  are  intermediate  between  those  of  /(a;)=0, 
BJncc  if  a,  6,  and  c  arc  roots  of  fix)—'),  in  the  order  of  their  niaguitudcs,  just  before  x  reaches^* 


Sturm's  theorem  and  method.  221 

Finally,  as  we  have  before  shown  that  as  x  passes  through  all  values  from 

—  00  to  4-  oo ,  there  can  be  no  change  in  any  of  the  functions  except  /(.i)  which 
will  affect  the  number  of  variations  and  permanences  in  the  signs  of  the  func- 
tions, there  is  only  one  variation  lost  when  x  passes  through  any  root  of  /(.*■).— 0. 

250.  Cor.  1. —  To  ascertain  the  number  of  real  roots  of  the  equa- 
tion f  (x)  =  0,  ice  substitute  in  f  (x),  f '(x),  f,(x),  i,{x) f„(x^),* 

—  00  for  X,  and  note  the  number  of  variations  of  sig^is.  Then  sub- 
stitute +  CO  for  X,  and  note  the  number  of  variations.  The  excess 
of  the  7iumber  of  variations  in  the  former  case  over  that  in  the  latter 
indicates  the  number  of  real  roots  of  the  equation. 

This  is  a  direct  consequence  of  the  proposition,  since  as  x  increases  from  — oo , 
there  is  no  change  in  the  number  of  variations  of  the  signs  of  the  functions  ex- 
cept when  X  passes  through  a  root ;  and  every  time  that  it  does  pass  through  a 
rrmt  one  variation  is  lost,  and  only  one.  But  in  passing  from  —  oo  to  +  x> ,  x 
l)asses  through  all  real  values.  Hence  the  excess  of  the  number  of  variations 
for  a;  =  —  00  over  the  number  for  a;  =  -+-  oo  is  equal  to  the  total  number  of 
real  roots. 

200*  Cor.  2. — To  ascertahi  how  many  real  roots  of  f  (x)  =  0  lie 
between  any  two  numbers  as  a  and  b,  substitute  the  less  of  the  two 
numbers  in  f(x),  i'{\),  f)(x),  f.(x),  etc.,  and  note  the  number  of  vari- 
ations of  siyns.  Then  substitute  the  greater  and  note  the  nxmiber  of 
variations.  The  excess  of  the  number  of  variations  in  the  former 
case  over  that  in  the  latter  indicates  the  number  of  real  roots  betioeen 
the  numbers  a  and  b. 

This  appears  from  the  proposition  in  the  same  manner  as  Cor.  1. 

201,  Sen. — Since  the  total  number  of  roots  of  an  equation  corresponds 
to  the  degree  of  the  equation  (234),  if  we  ascertain  as  above  the  number  of 
rml  roots  in  any  given  equation,  the  number  of  imaginary  roots  is  known  by 
implication. 

262,  JProb. — To  compute  the  numerical  values  of  f  (x),  f '(x), 
fj(x),  f2(x),  etc.^  i.e.,  of  any  function  of  x  for  any particxdar  value 
of  X,  when  the  function  is  of  the  form  Ax"  +  Bx""*  +  Cx""' 
4-  Dx"-' P. 

Solution. — Of  course  this  can  be  done  by  merely  substituting  the  proposed 
value  of  X  in  the  function.  But  there  is  a  more  elegant  and  expeditious  way, 
which  we  proceed  to  exhibit. 


root  rt,  /(J?)  and  /'(x)  have  different  signs,  and  just  ajttr,  they  have  like  signs.  But  just  before 
X  reaches  6,  /(x)  and  /'(r)  have  unlike  t?igns,  and  as  /(x)  cannot  have  changed  sign,  the  sign 
of  /'(r)  must  have  changed;  i.e.,  x  must  have  passed  through  a  root  of  /'(a;)=0,  in  passing 
from  a  to  h.  In  like  manner  it  may  be  shown  that  a  root  of  fm  lies  between  each  two  com- 
secntive  roots  of  /(j-)=0.    This  makes  f'{x)^^  have  one  root  les>^  than  /(a;)=0,  as  it  should. 

*  By  this  notation  Is  meant  the  wth  or  last  of  the  Stnrmian  Juncticms,  in  which  x  does  not 
appear;  or,  what  is  the  same  thintr,  that  in  which  the  exponent  of  x  \>  0. 


ADVANCED  COURSE  IN  ALGEBRA. 

Thus,  let  it  be  required  to  evaluate  Ax'^ -\- Bx* -\- Cx^  +  Bx^ -\- Ux  +  F  for 
X  =  a.  Multiply  A  hy  a  and  add  the  product  to  B.  Multiply  this  sum  by  a 
and  add  the  product  to  C.  Multiply  this  sum  by  a  and  add  the  product  to  J). 
Continue  this  operation  till  all  the  coefficients  have  been  involved  and  the  abso- 
lute term  added.  The  last  sum  is  the  value  of  the  function  when  a  is  substi- 
tuted for  X,  as  will  appear  from  considering  the  following : 


a 
Aa 

+  B 
a 

Aa* 

+  Ba+  ... 
a 

Aa' 

+  Ba*  jfVa  +  D 
a 

Aa* 

+  Ba*  +  C'a* 

-\-lJa 

+  B 
a 

Aa"-  +  Ba*  +  (Ja^  +  Ba*  -\-  Ea  +  F. 
This  is  evidently  the  value  of  the  function  when  a  is  substituted  for  x. 

N.  B. — 1.  If  the  function  is  not  complete,  i.  e.,  if  it  lacks  any  of  the  succes- 
sive powers  of  r,  rare  must  be  taken  to  supply  the  lacking  coefficients  with  O's. 
Thus  the  coefficients  of  x*  —  2j**  -+-  5  are  to  be  considered  aa  1 ,  0,  —  2,  0,  and 
5  (which  may  be  called  the  coefficient  of  a*"). 

2.  When  the  numbers  involved  are  small  the  operation  can  be  performed 
mentally. 

Ex.  1.  Evaluate  'ZbW  -  312a:»  +  1553x  -  5247865   for  x  =  342. 

OPERATION. 


175164 

350328 
262746 
29953044 
1553 
29954597 

342 

59909194 

119818388 

89803791 

10244472174 

-  524786.5 

10239224309        The  value  required. 


Sturm's  method.  223 

Ex.  2.  Evaluate   a^  —  Sx"  +  bx  —  20    for    x  =  2,  performing  the 
operation  mentally. 

Examples  of  the  Use  of  Sturm's  Method. 

1.  Find  the  number  and  situation  of  the  real  roots  of  a;'  —  4:X* 
-  6a;  +  8  =  0. 

Sug's. — If  the  student  has  attended  carefully  to  what  precedes,  he  will  have 
no  difficulty  in  determining  that 

f(x)  =  x^  —  4x^  -Qx  +  S; 
fix)  =  Sx^  -Sx-Q; 
/.(rc)  =  172^-12; 
and  /j(a;")  =  1467. 

Now,  for  ;r  =  -  00 ,  we  have /(a?)  -,  f'(x)  +,  /,(.t)  -,  and  fi(x^)  +;  i.  c.,the 
signs  of  the  functions  are 1 h.     There  are  therefore  three  variations. 

Again,  when  x=  +  oo ,  the  signs  are  +  +  +  +,  giving  no  variations.     Hence 
the  number  of  real  roots  is  3  —  0  =  3 ;  i.  e.,  they  are  all  real. 

To  find  the  situation  of  tfiese  roots  we  observe  that  for  x  =  0,  the  signs  of  the 

functions  are   H h,  giving  two  variations,  or  one  less  than  —  c-  gives. 

Hence  there  is  one  root  between  —  oo  and  0 ;  i.  e.,  one  negative  root.  The  other 
two  must  of  course  be  positive.  We  will  first  seek  the  situation  of  this  negative 
root.    Evaluate  by  {2G2). 

For    X  =  0,         the  signs  of  the  functions  are     H 1-. 

"       x=  -\,       «        "  "  "  "       -f.  4-  _  4-. 

"       a;  =  -  2,      "       "         "  "  "      -  H +.* 

Hence,  as  one  variation  is  lost  when  x  passes  from  —  2  to  —  1,  there  is  one  root 
between  —  1  and  —  2 ;  i.  c,  the  negative  root  is  —  1  and  a  fraction. 

In  like  manner  seeking  the  situation  of  the  positive  roots,  evaluating  the 
functions  by  (202),  we  have 

For    X  =  0,    the  signs     -\ (- ,  2  variations. 

'*      x=l,      "       " +  +,  1 

«      x  =  2,      "       " 4-  +,  1 

«      a;  =  3,      "       " +  +,  1 

"      a;  =  4,      "       "  —  +  +  +,  1 

"      .r  =  5,      "       "  -f-  +  +  +,  0 


*  The  evaluation  of  thcpc  functions  is  most  elegantly  and  oxpeditiouply  cfTected  by  (262). 
Thus  for  x=-2  we  have 

1        -4        -0        +  8  1^  3        -  S        -6  I   -2 

-2         Vi        -U  -  6         28 

-6  6        -  4=f(x)  -14         a2=/'(a;) 

When  the  value  of  x  for  which  we  arc  cvalnatina  is  small,  and  the  coefficients  also  small,  this 
process  can  be  carried  on  mentally  without  writing,  and  should  be  so  done. 


224  ADVANCED    COURSE   IN    AL(;EEKA. 

Therefore,  as  ono  variation  is  lost  wlien  x  passes  from  0  to  1 ,  there  is  one  root 
between  0  and  1,  ?*.  e.,  an  incommensurable  decimal.  Again,  one  variation  is  lost 
when  X  passes  from  4  to  5  ;  lience  the  other  root  lies  between  4  and  5,  or  is  4 
and  an  incommensurable  decimal. 

2(iS,  Sen.  2. — It  is  usually  unnecessary  to  find  fn{x")  (the  last  of  tlie 
Stunniun  functions),  since  its  sign,  which  is  all  that  is  important,  can  be 
detennined  by  inspection  from  tlie  next  to  the  last  function  and  the  pre- 
ceding divisor.  Thus,  if  we  were  to  divide  ar' -h  22.C  —  102  by  122x  —  393, 
first  multiplying  the  former  by  122,  it  would  be  clear  that  the  remainder 
would  be  —,  .without  going  through  the  operation.  Hence  /«(«")  would 
be  +. 

2  to  7.  Find  the  number  and  situation  of  the  real  roots  of  the 
following : 

(2.)  a:'  -f  6a:'  +  10a;  -  1  =  0;  (5.)  ar*  -  2a;^  +  a:^  -  8a;  +  6  =  0; 
(3.)  ar'  -  6a;'  +  8a;  4-  40  =  0;  (6.)  .a;^  -  4ar'  +  a;'  +  6a;  +  2  =  0; 
(4.)  a;*  _  4a;»  -  3a;  +  23  =  0;      (7.)  a;*  +  2a;'  +  17a;'  -  20a;  +  100  =  0. 

264,  ScH.  3. — In  case  the  equation  has  equal  roots,  we  shall  detect  them 
in  the  process  of  producing  the  Stumiian  functions,  since  in  such  a  case  the 
division  will  become  exact  at  some  stage  of  the  process,  and  the  last  Stur- 
mian  function  will  be  0.  Having  thus  discovered  that  the  equation  has 
equal  roots,  we  might  divide  out  the  factors  containing  them,  and  then  ope- 
rate on  the  depressed  equation  as  above  for  the  unequal  roots.  But  it  is 
not  necessary  to  depress  the  degree  of  the  equation,  since  the  several  Stur- 
mian  functions  will  have  the  same  variations  of  signs  in  either  case  for  any 
particular  value  of  a*.  This  arises  from  the  fact  that  the  common  divisor  of 
f{x)  and  f'{x)y  which  contains  the  equal  roots,  is  a  factor  of  each  of  the 
Sturmian  functions,  and  hence  its  presence  or  absence  will  not  affect  their 
signs  for  any  particular  value  of  x  if  the  common  factor  is  -H  for  this  value, 
and  will  change  the  signs  of  all  if  it  is  — ;  but  in  either  case  the  variations 
of  signs  will  not  be  affected. 

8.  Find  the  number  and  situation  of  the  unequal  real  roots  of 
ar*  —  6a;*  -f  7.r*  +  22a;'  —  GOx  -f-  40  =  0,  without  depressing  the  equa- 
tion. 

Bug's. — Forming  the  required  functions,  we  have 

f{x)  =  ar»  -  (xc*  +  Zv^  +  22j;*  -  OO.r  +  40 ; 
/  '{x)  =  5x*  -  24e=»  H-  21«*  +  44c  -  CO ; 
fi{x)  =  dlx^  -  228aJ«  +  4C8a;  -  320 ; 
fi{x)  =  a;*  -  4a?  +  4  ; 

Now  fii^x)  is  a  factor  of  f{x),  /'(«),  and  fi{x),  and  removing  it  from  fUl,  we 


STURM'S  METHOD.  225 

shall  have  the  following  functions,  which  nxay  be  used  instead  of  the  Sturmian 
functions  derived  from  the  depressed  equation : 

f\a:)  =  6x''  -  4x-15; 
/,(«)  =  37a; -80; 

Hence,  since  the  signs  of  these  two  sets  of  functions  evaluated  for  any  particular 
value  of  X  will  be  the  same,  either  set  may  be  used  at  pleasure. 

Thus  either  set  gives 

For    X  z=  —  CO , 1 \-; 

and  for   a*  =  +  00 ,     +  -f-  +  -)_. 

Therefore  there  are  two  unequal  real  roots  of  f{x)  =  0 ;  and  from  the  existence 
of  the  factor  {x  —2)*  mf{x)  and /'(a;),  we  know  that  there  are  three  equal  roots, 
each  2. 

The  situation  of  the  unequal  roots  can  now  be  found  as  before. 

9  to  12.  Find  th6  number  and  situation  of  the  real  roots  of  the 
following : 

(9.)  x^  -  "^x*  +  ISar'  +  Ux'  _  66a;  +  72  =  0; 
(10.)  a^  -  ISx"  -  28a;'^  +  2^x  -f  48  =  0; 
(11.)  a^  -  4.0^  -h  x'  +  'ZOx  +  Id  =  0; 
(12.)  a^  -  lOa^  +  6x  -{-  1  =  0. 

265,  ScH.  4. — Elegant  as  the  method  of  Sturm  is,  and  perfectly  as  it 
accomplishes  its  object,  the  labor  of  producing  the  functions  required  and 
evaluating  them,  especially  wlien  the  roots  are  large  and  widely  separated, 
is  so  great  as  to  deter  us  from  its  use  when  less  laborious  methods  will  serve 
the  purpose.  In  a  great  majority  of  practical  cases  in  which  there  are  no  equal 
roots^  the  2>rinciple  that  f  (x)  changes  sign  when  x  passes  through  a  root  of  f  (x)  =  0 
wiU  enahle  ns  to  determine  the  situation  of  the  roots  with  far  less  bdior  than  Stui^i^s 
Theorem.  Often  a  simple  inspection  of  the  equation  will  determine  the  near 
value  of  a  root.  Methods  are  usually  given  for  ascertaining  the  limits  (as 
they  are  improperly  called)  of  the  roots  of  an  equation,  from  the  coefficients. 
But  these  are  of  little  practical  value.* 

♦  For  example,  the  two  following,  which  are  most  frequently  given : 

1.  In  any  equation  the  greatest  negative  coefficient  •i'Vh  Hs  dan  changed  and  ina^ased  by  unity 
is  a  auPERion  limit  of  (he  roots. 

2.  In  any  equation  unity  added  to  that  root  of  the  greatest  v>egatlve  coefficient  with  its  sign 
changed,  whose  index  is  equal  to  the  difference  of  the  expomnts  of  the  first  term,  and  the  first  nega- 
tive term  is  a  pupkuior  limit. 

Now  consider  tho  equation  x''^  j-  a?' -500=0.  By  the  fir.-'t  rule  the  superior  limit  of  a  root  is 
501,  and  by  the  second  v/50b  + 1,  or  23  +  .  Now  the  fact  is,  the  greatest  root  is  7.6  +  .  Again,  by 
1,  the  superior  limit  of  the  roots  of  a;*  -3a;«  -48x-72=0  is  73 ;  and  by  2  it  is  the  same.  But  the 
greatest  root  is  9. 


4-1 

7 

0 
56 

-500 
392 

ij: 

8 

56 

-103, 

i.e.,    fix)  is-. 

+1 
8 

0 

72 

-500  j 
576 

_8 

9 

72 

76, 

i.e.,  /(aj)is-f. 

22*5  ADVANCED   COURSE    IN   ALGKBIIA. 

13.  Find  by  inspection,  and  also  by  Sturm's  method,  ilie  situation 
of  tiie  roots  of  the  equation  a:*  +  a;*  —  500  =  0. 

Sug's. — Ijet  the  student  apply  Sturm's  method.    The  following  is  a  solution 
by  inspection : 

3 3. 

Since  x  =  r  500  —  x^,  there  is  a  +  root  less  than  r  500,  or  less  than  8.   Now, 
trying  7,  we  have 


Trying  8, 


There  is  therefore  a  root  between  7  and  8. 

Also  from  the  "elation  t  =  r  500— .«*,  or  from  the  operations  above,  we  see 
that  there  is  no  other  positive  root ;  since  fix)  evaluated  for  any  positive  quan- 
tity less  than  7  would  certainly  be  — ,  and  for  anything  greater  than  8,  -f. 

Finally,  that  there  can  be  no  negative  root  is  evident,  since  r  500— «'  cannot 

be  negative  until  sc'  >  500,  but  then  r 500— «•  <  y—x*,  and  v—x*  is  always 

<  X.      Hence  for  x  negative   we  can  never  have  x  =  y  500  —  «*.     Tlierefore 
our  equation  has  one  real  and  two  imaginary  roots. 

Note.— The  advantage  of  this  method  of  inspection  over  Sturm's  method,  in 
this  case,  will  not  be  fully  seen  unless  the  student  observes  that  all  this  can  be 
done  mentally,  without  writing  a  single  figure. 

14.  Find  by  inspection,  and  also  by  Sturm's  method,  the  number 
and  situation  of  the  real  roots  of  a:'  +  a;'  +  rr  —  100  =  0. 

Suo's. — A  mere  glance  should  show  that  there  can  be  but  one  positive  root, 
and  that  that  is  less  than  5.  In  like  manner  writing  x-^  —  x*  +  x  +  100  =  0,  or 
X*  +  X  +  100  =  ar*,  we  see  that  no  positive  value  of  x  can  satisfy  the  equation; 
for  when  x  is  less  than  1,  of  course  the  first  member  is  greater  than  the  second, 
and  when  x  is  greater  than  1,  x*  itself  is  greater  than  «*. 

15.  Find,  by  inspecting  the  changes  of  sign  of  f(x)  for  varying 
viilues  of  X,  the  situation  of  the  roots  of  ar'  —  3:r  —  1  =  0,  and  also 
by  Sturm's  method. 

16.  Find  by  inspection  the  situation  of  the  roots  of  ar'  —  22a; 
-  24  =  0. 

Sug's. — Writing  x{x^  —  22)  =  24,  we  see  that  any  positive  value  of  x  which 
satisfies  this  must  make  x^  >  22,  that  is,  must  be  greater  than  4.  But  5  makes 
x(x*  —  22)  =  15,  and  6  makes  it  84.     Moreover,  it  is  evident  that  no  number 


STURM'S   METHOD.  227 

greater  than  C  will  satisfy  the  o<]  nation.  Seeking  for  negative  roots,  we  write 
x^-22x  +  24:  =  0;  and  then  x{.ir  -  22)  =  -  24.  To  satisfy  this,  x^  must  be  less 
than  22,  or  ^  <  5.  For  x  =  0,  f(x)  is  +;  for  «  =  1,  /(x)  is  +;  for  a;  =  2,  /(.r) 
is  — .  Hence  a  root  of  the  given  equation  between  —  1  and  —2.  Finally,  for 
x  —  'd,  /(.r)  is  — ;  but  for  x  =  4,  f{x)  =  0.  Hence  a  root  of  the  given  equation 
is  —4. 

17.  Determine  the  situation  of  the  roots  of  x'—  10a;' -f-  C)x  +  1  =  0, 
by  examining  the  changes  of  sign  of  f{x). 

SuG's.— For  a;  r=  0,  f(x)  is  +;  for  a;  =  1,  f{x)  is  -;  for  a;  =  2,  f{x)  is  -;  for 
^  —  'S,  f{x)ia  — ;  for  a;  =  4,  f(x)  is  +;  and  will  evidently  remain  +,  as  x  ad- 
vances beyond  4.    This  is  seen  from  the  following  : 

1        0        -10  0        +6        +1  I  4 

4  16        24        +96        408 

4        ^        24  102        409 

Now  any  positive  number  greater  than  4  would  destroy  the  —10  in  this  pro- 
cess, and  give  the  sum  at  that  point  greater  than  6,  and  hence  the  aggregate 
would  rapidly  increase.    Thus  notice,  when  3  is  substituted,  we  have 
1        0        -10  0  +0        +1  |_3^ 

n  9       -3  -9         -9 

3        _  1       _3  __:]        _8 

Now  3  is  not  large  enough  to  destroy  the  —10;  but  every  number  larger  than 
4  will  destroy  it. 

To  examine  for  negative  roots  we  write  x^—  10a; '4-  6a;  —  1  =  0.  In  this,  for 
x  =  0,  f{x)  is  -;  for  x=  1,  f(x)  is  — ;  for  x  =  2,  f{x)  is  -;  for  a;  =  3,  /(a-)  is 
— ;  but  for  a;  =  4,  and  all  numbers  greater  than  4,/(.r)  is  +. 

We  have  now  found  that  there  are  certainly  three  roots  between  —  4  and 
-4-  4,  and  none  beyond  these  limits  either  way.  But  it  is  not  safe  to  conclude  that 
the  other  tioo  roots  are  imaginary.  The  fact  is,  they  are  not.  How,  then,  are  we 
to  find  them  ?  Sturm's  method  is  thought  to  possess  particular  advantage  in 
saving  us  from  such  erroneous  conclusions,  and  enabling  us  to  find  the  eituation 
of  all  the  real  roots  with  infallible  certainty.  And  certainly  it  does  do  this ;  but 
let  us  see  if  we  cannot  do  it,  in  this  instance  at  least,  as  readily  without  that 
method.  It  will  be  observed  that  wo  know  only  that  —  3  is  the  initial  figure 
of  one  root,  and  +  3  of  another.  The  initial  digit  of  the  root  between 
0  and  -(-  1  we  have  not  found.  Let  us  seek  it.  For  a;  =  0,  f{x)  is  + ;  and 
by  trying  a;  =  .1,  x  =  .2,  we  should  at  once  see  that /(a;)  changes  very  slowly, 
and  as  when  a;  =  1,  f{x)  is  only  —  2,  we  should  be  led  to  try  numbers  near  1. 
Trying  x  =.8,  we  would  find  that  f{x)  is  +,  and  for  x  =.9, /(a?)  is  — .  Hence  .8 
is  the  initial  figure  of  the  root  lying  between  0  and  +  1. 

We  now  know  the  initial  figures  of  three  of  the  roots.  But  where  are  the 
other  two  roots  ?  If  they  are  real  we  know  that  they  lie  between  —  4  and  +  4. 
as  we  have  seen  above  that  no  root  can  lie  beyond  these  limits.  Moreover,  aa 
the  function  changes  value  rapidly  beyond  1,  and  slowly  between  —  1  and  1.  i« 


228  ADVANCED  COURSE  IN  ALGEBRA. 

would  naturally  Ik^  suggested  that  there  may  be  two  changes  of  sign  between  0 
and  +  1,  or  0  and  —  1.  Evaluating  f{x)  =  x'^  —  10^ '  -t-  Oc  +  1  for  .1,  .2,  .3,  etc.. 
we  soon  see  that  it  will  not  change  sign  for  values  of  x  between  0  and  +  i. 
Evaluating  f{x)  =  x^  —  10.f'  +  ftr  —  1  for  .1,  .2,  .3,  etc.,  we  find  that  the  other 
roots  are  between  0  and  —  1,  and  that  their  initial  digits  are  —.1  and  —.6. 

18  to  23.  Find  by  inspection,  by  the  change  in  sign  of  /(x),  or  by 
Sturm's  method,  the  number  and  situation  of  the  real  roots  of  the 
following : 

(18.)  .r'-  ^x'-  4tx  +  11  =  0; 

(19.)  ar'-2a:-5  =  0; 

(20.)  X*  -  4.r'  -  3.r  +  23  =  0 ; 

(21.)  .r''  +  11a;'  -  102.c  +  181  =  0; 

(22.)  x'-llx'  -^  54.r  =  350  ; 

(23.)  x^  +  2x'  +  3.r  -  13089030  =  0.* 

206.  Sen.  5. — If  we  have  an  equation  in  which,  when  cleared  of  frac- 
tions, the  coefficient  of  the  highest  power  of  ac  is  not  unity,  it  may  be  trans- 
formed by  {22S)  into  one  having  such  coefficient.  But  thi^  it  not  necessary 
in  order  to  the  application  of  Sturm's  method^  as  it  is  not  required  by  anything  in 
the  demonstration  of  that  theorem  that  the  coefficients  should  be  integral, 

24  to  31.  Find  l)y  Sturm's  method  the  number  and  situutiou  of 
the  real  roots  of  the  following: 

(24.)  2.r'  +  3a:'-  4a:  -  10  =  0 ;  (28.)  3.*:*-  4.c'  +  2x  -  1000  r^  0 ; 

(25.)  x"-  18^.c  +  20,5^3^  =  0 ;  t         (29.)  Ix'-  83:c  +  187  =  0 ; 
(26.)  %3^-  36a:'  +  4G.c  -15  =  0;        (30.)  x"-  lf.6'-  Ifa:  =  440 ; 
(27.)  4a:'-  12a:'  -h  1  l.r  -  3  =  0  ;  (31.)  x^-  \x'-  |a:  =  312. 


Horner's  Method  of  Solution.^ 

207,  Horner's  method  of  solving  numerical  equations  is  a  method 
of  finding  the  incommensurable  roots  of  such  equations  to  any  re- 

•  Observe  that  neglecting  the  termei  2a:'  +  Sx,  which,  since  x  is  larjjc,  arc  small  as  compared 
with  x',  we  liavu  x3  =  1.3089030,  or  x  lics>  between  200  and  20k)  probab'y. 

t  Clear  of  fractions  first. 

X  Among  the  manj  methodp  dis^covcred.  and  ("loiibtlc.-s  to  be  discovered,  for  (his  purpose,  it 
is  scarcely  pos?-il)le  that  Homer's  should  be  Hiperceded,  since  the  solution  of  pucli  an  equation 
will  certainly  require  the  extraction  of  a  root  corresponding  to  the  defrree  of  the  equation  ;  iind 
the  labor  required  by  Horner's  method  is  not  greater  than  that  required  lo  extract  this  root. 
Nor  is  this  merely  a  method  of  approximation,  except  as  any  mcthfti  for  inrommensurafjle  roots 
is  necessarily  a  method  of  approximation.  If  tlie  nK>t  can  be  expressed  exactly  in  the  decimal 
notation,  or  by  means  of  a  repealing  decimal,  this  process  tffects  it.  The  method  was  first 
published  by  W.  G.  H<»rncr.  Etq.,  «  f  Bath,  England,  in  1819,  about  fifteen  years  before  Sturm'* 
Theorem  wis  published. 


HOKNEIIS   METHOD. 

quired  degree  of  approximate  accuracy.     It  is  based  upon  the  two 
following  problems  and  proposition  : 

208,  JProb, — 7h  transform  mi  equation,  as  f  (x)  =  0,  into  another 
whose  roots  shall  be  a  less  than  those  of  tJie  given  equation. 

Solution. — Let  x=a-\-Xi,  whence  a?,  —x—a,  and  we  hare  f{x)=f{a+Xi)=0, 
or  0=f{a  +  Xi).      Deyeloping    the    latter    by    Taylor's    Formula,    we    have 

0=f{a  +  x,)  =  fia)+ f'{a)x^+f"{a)~  +   /"'{«)p  +  /*M«)^  +  etc..  or 

0  =/(«)  +f'{a)Xx  +f"{a)'^  +/'"(«)  j3  +/''^(«)  i^  >  etc.,  as  the  required  equa- 
tion. 

209,  Sen. — The  meaning  of  this  may  be  stated  thus  :  The  absolute  term 
of  the  transformed  equation  is  the  value  of  /(r)  when  a  is  substituted  for 
a?;  the  coefficient  of  the  first  power  of  the  unknown  quantity,  ;r,,  in  the 
new  equation  is  the  first  differential  coefficient  of  f{x),  when  a  is  substituted  for 
X  in  this  coefficient ;  the  coefficient  of  the  second  power  of  X\  is  ^  the  second  dif 
ferential  coefficient  of  f(x),  when  a  is  substituted  for  x ;  etc. 

Ex. — From  5x*  —  12.c^  +  3aj'  +  4x  +  5  =  0  deduce  a  new  equation 
whose  roots  shall  be  each  less  by  2  than  the  roots  of  this. 

SOLUTION. 

f(x)  =  5x*  -  nx^  +  3a;2  +  4a?  +  5  =9  =f{a). 

x=a=2* 

f\x)  =  20a;'  -  C().r '  +  G.c  +  4        =83  =f\a). 

x=Z 

f%r)  =  60x'  -  73  r  +  6        =102  =f"a.   /.  \f'\a)  =  51. 
a:=2 

f'"(x)  =  120^  -  73        =108  =f"\a).        .'.  X  f"'{a)  =  28. 
f^r^T)  =  130        =  120  =/•%).  .'.  ,-i  /'%)  =  6. 

x=2 

Hence  0  =  0  +  JJ3r,  +  51.?,*+  2a»t'+  5.p,\  or  Sv,^  +  28.c,^-t-51a;i''+32a;i+  0  =  0, 
is  an  equation  wluwc  roots  are  2  less  than  the  roots  of  the  given  equation, 
since  a;,  =  j;  —  2. 

270*  IPvob, —  To  compute  the  numerical  values  of  f  (a),  f '(u), 
if  "(a),  jj-f '"(a),  jf  f^(a),  etc.,  from  f(x),  when  f(x)  has  the  form 
Ax"+  Bx"-'  -f  Cx'"-'+  Bx"-^ P. 

Solution -—Let  /(.t)  =  Ax^+  Bx^+  Cx^  -i-  Dx  +  E;  whence,  forming  f'{x), 
f'\x),  f"'{x),  and  /""(.f),  and  substituting  a  for  x,  we  have 


♦  TTi"  meaning  of  lhi»  Hotatlon  ii»  Ihat  X  i*  made  eqnal  to  2  In  the  function  whence  rei^«lt» 
the  following  value. 


2ae 


ADVANCED  COURSE  IN  ALGEBRA. 


/(«)  =  Aa*  +  Ba^  +  Ca^  +  Da  +  E; 
f'(a)  =  4Aa^  +  3Ba'  +  2Ca  +  I) ; 
if"{a)  =  QAa^  +  'dBa+  C; 
^hf"'{a)  =  ^Aa  +  B; 
,i/-(a)  =  A. 


Now,  we  may  compute  these  as  follows 


+   + 

w 


c^ 

+ 

+ 

M 

1 

5 

3 

+ 

+ 

+ 

•4 

M 

•« 

e 

« 

e 

^ 

O) 

+ 

+ 

rt 

•0 

« 

e 

^ 

CO 

t 

H 


3 
k 


6  .-S  6 


3  !».  « 


Si  = 

5 -a '5 

^    ^    be 

ca 


"^  ''^  p  s 

c  -  =  "^ 

c  ^  -^  '^ 

C  ac  a 


'^  ^ 


'^  z: 
'§  % 


I  i  2 
I  ^  S 

s  S;'i 


--  .2  H 

H    '3   r.^ 

fi    o    s 

C-'S 


ho 


■^  S5 


■a 

3 


c 
o 

^      §     S 

I  I  i' 


7   ® 
+   c 


O     +     cS 


ri     '^       O      .^      ♦* 


be  5 

I-? 


0 


O    n 

XI    o 


Examples. 

1.  Transform  3a^— 42:*  +  Ta;*^  +  8a:— 12=0  into  another,  equation 
each  of  whose  roots  shall  be  3  less  than  the  roots  of  this. 


Solution. — Arranging  the  coeflBcients  and  proceeding  as  in  the  above  solu 
tion,  we  ha   i  the  following  : 


OPERATION. 

+  7     +8 

-12 

1  3 

15      66 

222 

22      74 

210  = 

=  /(3) 

42     192 

64     266  = 

=  /'(3) 

69 

133  =  i/"(3) 

Horner's  method.  231 


s       -4 
_9 

5 

_9 
14 

Jl 

23 
_9 

32  =  \hf"'{S). 

Hence  the  transformed  equation  is 

dxi^+^zi'  +  133a;  ,2  +  266a;,  +210  =  0. 

2.  Transform  Sa^  —  13.^'  + 7a:'  —  8a:  —  9  =  0  into  another  equation 
wliose  roots  shall  be  less  by  3  than  the  roots  of  this. 

The  new  equation  is  3a:' +  23^'"  + 52a:'' +  7a:— 78=0.* 

3.  Transform  x^  +  2x''  —  6a:''  —  10a:  +  8  =  0  into  another  equation 
whose  roots  shall  be  2  less  than  the  roots  of  this. 


I  2 


PROCESS. 

0 

+  2 

-6 

-10 

+  8 

2 

4 

12 

12 

4 

2 

6 

6 

2 

12 

2 

8 

28 

68 

4 

14 

34 

70 

2 

12 

52 

6 

26 

86 

2 

i? 

8 

42 

_2 

10 
.'.  The  equation  is  aj"  +  10a;*  +  42a;^  +  86a;*  +  70a;  + 12=0. 

4.  Transform  x'  -  Gx"  +  7.4a:'  +  7.92a:'  -  17.872a:-.79232  =  0  into 
another  equation  whose  roots  shall  be  each  less  by  1.2  than  the  roots 
of  this. 

5.  Transform  a:^— 2a:' +  3a: +  4=0  into  another  equation  whose  roots 
shall  be  1.7  less  than  the  roots  of  this. 

6.  Transform  a^H  Ha:'— 102a:  + 181=0  into  another  equation  whose 
roots  shall  he  3  less  than  the  roots  of  this  equation  :  transform  the 

♦  For  convenience  in  reading  and  writing,  it  is  customary  to  omit  the  subecripts  which  dis- 
llnjfuis'h  <hc  nnlcnown  quantity  in  the  transformed  equation  from  that  in  the  given  equation. 
But  it  should  be  borne  in  mind  that  the  iinlcnown  quantities  are  different. 


232 


ADVANCED   COUllSE  IN  ALGEBRA. 


resulting  equation  into  another  whose  roots  shall  be  .2  less  than  the 
roots  of  the  last:  transform  this  equation  into  another  whose  roots 
shall  be  .01  less  than  those  of  the  last :  transform  this  into  another 
whose  roots  shall  be  .003  less  than  its  roots. 


OPERATION. 


+  11 

14 

_8 

17 

3 

20* 
.2 

ao.2 

20.4 

ao.6t 

.01 

20.61 

.01 

20.G3 
.01 

20.63t 
.008 


-102 

J? 

-60 

_51 

4.04 

-4.9G 

4.08 


+  181 
-180 


1* 
-.992 

.098+ 
-.000739 

.001201$ 
-.001217403 


.01 


.003 


.2061 
.6739 


-.4677  J 
.061899 

-.405801 
.061908 
-.343893^ 


EXPLANATION. 

*  These,  together  with  the  first, 
are  the  coefficienta  of  the  cquiition 
whose  roots  are  3  less  tlum  those 
of  the  given  equation.  The  equa- 
tion written  out  is  a;'  +  20x*— 9.C 
+  1=0.  (.1).  But,  instead  of  re- 
writing these  coefficients  for  the 
second  transformation,  we  operate  upon  them  just  as 
they  stand. 

f  These,  together  with  the  first,  are  the  coefficients 
of  the  equation  whose  roots  arc  .2  less  than  those  of 
{A),  and  consequently  3.2  less  than  those  of  the  given 
equation.     This  equation  written  out  is  :v*  +  20.(yX'- ~ 
.88.C+. 008=0.  {B).     But  instead  of  rewriting  these  co- 
efficients we  effect  the  next  transformation  upon  them 
just  as  they  stand. 
I  These,  together  with  the  first  (which  remains  the  same  in  all),  are  the  co- 
efficients of  the  e(iuation  whose  roots  are  .01  less  than  the  roots  of  (B),  .21  less 
than  the  roots  of  {A),  and  3.21  less  than  the  roots  of  the  given  equation.     This 
equation  is  a:'  +20.63J;-  -.4677.r  +  .001261=0.  (C). 

^  These  are  the  coefficients  of  the  equation  whose  roots  are  .OOo  Ics.'J  than 
those  of  {€),  .013  less  than  those  of  {B),  .213  less  than  those  of  {A),  and  3.213 
less  than  those  of  the  given  equation.  The  last  transformed  equation  is 
a?  •  +  20.639^;'  -  .343893.C  +  .000043597=0. 

7.  Transform,  as  above,  the  equation  x^—l^x^-hVix—S  —  O,  suc- 
cessively, into  equations  whose  roots  shall  be  2  less,  2.8  less,  and  2.85 
less  than  the  roots  of  the  given  equation. 


HORNER  S   METHOD. 


233 


6 
_2 

8» 

8.8 
_.S 
9.6 

10.4 
.8 
11.2t 

m 


OPERATION. 

-13 

+  12 

-3               |! 

4 

-16 

-8 

-8 

-4 

-11* 

8 

0 

8.9856 

0 

-4* 

-2.0144t 

12 

15.232 

1.71940625 

12* 

11.232 

-.294993751 

7.04 

21.376 

19.04 

32.608t 

7.68 

1.780125 

26.72 

84.388125 

8.32 

l.e08375 

35.04f 

o6.1'JG500t 

.5625 

35.6025 

.5650 

J:6.1675 

.5675 

I  2.85 


36.7850$ 


{B). 


11.25 
.05 

11.30 
.05 
ll.;i5 
.ft5 
11.40t 
Hence  the  successive  equations  are. 
The  Primitive,  ^^  -12.c«  +  12a;  -3-0  ; 

One  whose  roots  are  2  less  than  those  of  (.4), 

ar^+8c'  +-12.c^-4,c-ll=0 
One  whose  roots  are  .8  less  than  those  of  {B),  or  2.8  than  those  of  {A), 

X*  +  11.2.C '  +  35.04c2  +  32.608.^-2.0144=0 ;    {C). 
One  whose  roots  are  .05  less  than  those  of  (C),  .85  less  than  those  of  {B),  oi 
2.85  less  than  those  of  (.4), 

X*  +  11.4ar'  +  36.735x2  +  36.1965a;-.29499375=0. 

8.  Transform,  as  above,  the  equation  a.-^— 7a; +  7=0,  successively, 
into  equations  whose  roots  shall  be  1  less,  1.3  less,  1.35  less,  and  1.356 
less  than  the  roots  of  the  given  equation. 

271,  JPvop, — l/'a  +  Xi  is  a  root  of  /*(ic)=0,  andxx  is  sufficiently 


small  with  reference  to  a,  x^ 


/(«) 


,  approximately. 


234  ADVANCED  COUr^SE  IN  ALGEBRA. 

Dem. — If  a+Xi  is  a  root  of  /  (x)  =0,  f{a  +  a; , ) = 0.  Developing  tliis  by  Taylor's 
Fbrmola,  we  have 

/{a  +  x,)=f{a)+f(a)x,+f"{a)^  +  f"(a)^-^  etc.=0. 

Now,  to  determine  a?,  approximately,  which  is  all  the  proposition  proposes,  when 
Xx  is  quite  small  with  reference  to  a,  all  the  terms  in  the  development  involving 
higher  powers  of  a?  I  than  the  first  maybe  neglected;  whence  we  have  /(a)  4- 

/■(a)^,=0.or«,  =  --^j. 

Ex. — Knowing  that  4.+  some  decimal  fraction  which  we  will  call 
Xx  is  a  root  of  ar''  +  a;'  +  a;  — 100=0,  required  the  approximate  value  of 
the  decimal  fmction  x^. 

Solution. — Finding /(a),  i.e.,  in  this  case/ (4)*  in  the  ordinary  way,  we  have 

1         +1  +1  -100  |_4_ 

_4  JO  _84 

6  21  -16=/(a),or/(4)* 

4  _36 
9  57  =/'{«),  or/'(4)» 

_4 
13 

Hence  —  ——-•= -— -  =.28+  \^  approximately  the  decimal  part  of  the  root. 

/  \^)  o7 

In  fact,  2  is  the  tenths  figure  of  the  decimal  part  of  the  root,  the  root  being 

(as  we  shall  find  hereafter)  4.2644  + . 

We  thus  have  a^,*+  13^*,'+  57a?i  —16=0,  an  equation  whose  roots  are  4  less 

than  the  roots  of  the  given  equation.    We  will  now  transform  this  into  another 

equation  whose  roots  shall  be  .2  less  than  the  roots  of  this  equation,  or  4.2  less 

than  the  roots  of  the  given  equation.     Thus 


1 

+  13 

+  57 

-16 

1  -2 

.2 

2.64 

11.928 

13.2 

59.64 

-4.072  = 

=  /(4.2)t 

.2 

2.68 

13.4 

62.32  : 

=/'(4.2)t 

2 

13.6 

and  the  transformed  equation 

lis 

a.,»  + 

13.6.rj«  +  62.32«i -4.072=0, 

*  This  notation  means,  the  valne  ot/(x)  when  4  ie  snbstitnted  for  x  therein. 

t  That  these  are  the  values  of  f(x)  (the  first  member  of  the  {jiven  eqnation)  and/'(a;),  when 
4.'2  i!«  substituted  for  x,  will  be  evident  if  it  is  considered  that  they  are  the  same  results  as 
would  have  been  obtained  by  transforming  the  given  equation  immediately  (by  one  procei^e) 
into  another  whose  roots  are  4.2  less. 


Horner's  method.  285 

which  is  an  equation  whose  roots  are  4.3  less  than  those  of  the  given  equation, 
Hence  by  the  proposition  (Tg  =  —  "looo  —065  approximately.    In  fact,  it  will 

O/ft.O,* 

be  seen  that  6  is  the  hundredths  figure  of  the  root. 

Writing  both  portions  of  the  above  work  together,  it  stands  thus  : 


+1 

+  1 

-100 

[4.2 

4 

20 

84 

5 
4 

21 
36 

-16* 
11.928 

*.-. -16=/(«).or/{4) 

9 
4 

13* 
.2 

57* 
2.64 

59.64 
2.68 

-4.072f 

*  .'.   57=/'(«),  or/(4) 
t  .-.  -4.072=/(4.2) 

13.2 
.2 

13.4 

62.32t 

f.-.  62.32=/'(4.2) 

.2 

13.6t 

HoRiq^ER's  Rule. 

272.  RULE. — 1.  Put  the  equatioi?^  i^  the  for^i 
Ax^  +  Bsf-^  +  6'af-* Mx  +  L=Oy 

1^  WHICH  the  coefficients  A,  B,  C X,  IF    NOT   INTEGRAL, 

are  expressed  exactly  in  decimal  fractions. 

2.  Find  the  number  and  situation  of  the  positive  real 
ROOTS  BY  Sturm's  Theorem,  determining  one  or  jiore  (usually' 
two)  of  the  initial  figures.  (See  Sen.  1.) 

3.  Write  the  coefficients  in  order  with  their  propei^ 
signs,  being  careful  to  supply  with  O's  the  places  of  co- 
efficients OF  missing  terms,  if  the  equation  is  not  complete. 
Taking  the  initial  figures  of  one  of  these  roots  as  thus 
found,  operate  on  these  coefficients  so  as  to  obtain  the  co- 
efficients OP  the  transformed  equation  whose  roots  shall 

BE  less  by  the  portion  OF  THIS  ROOT  ALREADY  FOUND. 

4.  Having  found  these  coefficients,  if  the  coefficient  of 
the  first  power  of  the  unknown  quantity  in  this  trans- 


236  ADVANCED   COURSE   IN   ALGEBRA. 

FORMED  EQUATION  AND  THE  ABSOLUTE  TERM,  /'(«)  AND  /(«),  HAVE 
UNLIKE  SIGNS,  DIVIDE  THE  LATTER  BY  THE  FORMER,  AND  THE  FIRST 
FIGURE  OF  THIS  QUOTIENT  WILL  BE  (APPROXIMATELY)  THE  NEXT 
FIGURE  OF  THE  ROOT.  (See  ScH.  2.)  If  THESE  FUNCTIONS  HAVE 
LIKE  SIGNS,  MORE  FIGURES  OF  THE  ROOT  MUST  BE  FOUND  BY  StURM'S 

Theorem  or  by  trial,  before  proceeding  to  apply  this  pro- 
cess OF  transformation. 

5.  Having  found  a  figure  of  the  root  by  dividing  f{a)  by 

/'(a),  ANNEX  it  to  the  ROOT  AND   OPERATE   ON  THE   COEFFICIENTS 

of  the  last  (transformed)  equation  as  they  stand,  to  pro- 
duce the  coefficients  of  the  next  transformed  equation,  i.  <?., 
the  one  whose  roots  shall  be  less  than  those  of  the  last, 
by  the  last  figure  of  the  root,  and  less  than  those  of  the 
given  equation  by'  the  entire  portion  of  the  root  now  found. 
Having  found  these  coefficients,  divide  the  absolute  term 

BY  the  coefficient  OF  THE  FIRST  POWER  OF  THE  UNKNOW|<^ 
QUANTITY,  IF  THEIR  SIGNS  ARE  UNLIKE,  AND  THE  FIRST  FIGURE 
OF  THIS  QUOTIENT  WILL  BE  (APPROXIMATELY)  THE  NEXT  FIGURE 
OF  THE  ROOT.  If  THESE  SIGNS  ARE  ALIKE,  THE  LAST  ASSUMED 
FIGURE    OF    THE   ROOT    IS    TOO    LARGE    AND    MUST    BE    DIMINISHED. 

(See  ScH.  3.) 
G.  Proceed  in  this  manner  until  the  root  is  obtained; 

OR,  IF  THE  ROOT  IS  INCOMMENSURABLE,  UNTIL  AS  MANY  FIGURES 
OF  THE  DECIMAL  FRACTION  ARE  OBTAINED  AS  ARE  DESIRED.  (See 
SCH.  4.) 

7.  In  LIKE  MANNER  ALL  THE  POSITIVE  REAL  ROOTS,  OR  THEIR  AP- 
PROXIMATE VALUES,  MAY  BE  FOUND.  To  OBTAIN  THE  NEGATIVE 
ROOTS,  CHANGE  THE  SIGNS  OF  ALL  THE  TERMS  CONTAINING  ODD  POW- 
ERS OF  THE  UNKNOWN  QUANTITY,  OR  ALL  OF  THOSE  CONTAINING  THE 
EVEN  POWERS  ;  OR,  IF  THE  EQUATION  IS  COMPLETE,  EACH  ALTERNATE 
SIGN,  AND  PROCEED  TO  FIND  THE  POSITIVE  ROOTS  OF  THIS  EQUATION 
AS  BEFORE.  ThE  VALUES  THUS  FOUND  WILL  BE  THE  NUMERICAL 
VALUES  OF  THE  NEGATIVE  ROOTS  (246)* 

This  rule  is  based  upon  previously  demonstrated  principles,  and  needs  no 
special  demonstration. 

273.  ScH.  1.— By  means  of  (244,  24=5)  we  can  usually  find  the  initial 
figure  or  figures  of  the  roots  with  less  labor  than  l>y  Sturirv  •  Tlicoreni. 


HOliNERS  METHOD.  237 

274,  Sen.  2.— Since  hy  {271)x,=  -  J^,  if  both /(«)  and  f\a)  have 

the  same  sign  at  amj  time^  this  quotient  will  be  — ,  and  hence  the  value 
thus  found  for  a-,  will  not  be  the  amount  to  be  added  (annexed)  to  the  por- 
tion of  the  root  already  found,  for  the  assumption  is  that  this  portion  is  1er» 
than  the  root  of  the  equation  which  we  are  seeking. 

27''>.  Sen.  3.— That  the  figure  of  the  root  found  by  dividing/(«)  by/'{«) 
is  liable  to  be  too  large  is  readily  seen  when  we  consider  that  instead  of 
/'(«)^i=  — /(^)  (in  Dem.  of  271)1  we  should  have,  if  no  terms  were 
omitted, 

/'(«>», +i/"(«)^,'  +  i/"'(r«).c,'+etc.  =  -/(«). 

Now  a  value  of  .r,  which  satisfies  the  former  may  evidently  be  quite  too 
large  to  satisfy  the  latter.  Thus  consider  .t;^  +  10.?^  -i-5j;— 2600=0.  Neglect- 
ing x^  and  lO-c*,  we  have  5ar=2600,  or a=520.  But  this  will  by  no  means 
satisfy  the  equation  when  x^  and  10a;*  are  not  neglected. 

Again,  the  figure  found  by  dividing/(a)  hy  f'{a)  may  be  too  small.  Thus, 
if  we  have  .c^  —  lO-c^ +12.^—3=0,  and  neglect  .c*',  and  — 12.6%  we  have  12.r-3 
=0,  or  x=\.  But  this  is  too  small  a  value  to  satisfy  the  equation,  since  for 
x=\,  — 12j;*  will  be  numerically  much  larger  than  x*^  and  hence  retaining 
these  terms  will  diminish  the  function,  thus  making  \  too  small  to  satisfy 
the  equation. 

270,  Sen.  4. — From  Sen.  2  it  appears  that  f{(i)  cannot  change  sign  in 
the  process  unless  f'{a)  also  changes  sign.  But  when  f{a)  changes  sign,  we 
know  by  (244)  that  we  have  passed  a  root  of  the  equation ;  if,  however,  f'{a) 
also  changes  at  t!ic  same  time,  our  work  may  still  be  right.  In  such  a  case 
there  are  two  roots  having  tlicir  initial  figure  or  figures  alike,  e.  </.,  one  may 
be  23.56  +  ,  and  tlie  other,  23.59  +  .  To  obtain  the  less  of  the  two  roots,  take 
the  largest  figure  which  will  not  cause  cither  f  (a)  or  f'(a)  to  change  sign; 
and  for  the  larrrcr  of  the  two  roots  take  the  smallest  figure  which  will  cause 
both  f(a)  and  f\a)  to  change  sign. 

[Note. — Those  scholiums,  as  also  the  rule,  will  be  better  understood  in  con- 
nection with  their  applications  in  the  following  examples.  But  in  review,  after 
the  solution  of  the  examples,  they  should  bo  carefully  learned.] 


Examples. 

1.  Kequired  the  roots  of. r''—4.r-  —  C.r-f8=0. 

Soi.iTTiox. — By  Sturm's  method  wo  find  that  there  are  3  real  roots,  one  nega- 
tive, and  two  positive  (see  Ex.  1,  page  223),  and  also  that  the  negative  root  is 
—  1.  and  an  incommensurable  decimal,  that  one  positive  root  is  an  incommen- 
surable decimal,  and  that  the  other  positive  root  is  4.  +  an  incommensurable 
decimal.     We  will  seek  the  latter  first. 


238  ADVANCED   COUltSE   IN   ALGEBRA. 


-4  -6  ^-  8  I  4.892+ 

_4 

0 
_4 

4 

4 

8.8 
.8 

9.6 
.8 


10.49 
.00 

10.58 
.09 

10.673 
.002 

10.674 
.003 

10.676 


UfKKATlON. 

-6 

^-  8 

0 

-24 

-6 

-16...' 

16 

13.632 

10.. 

-2.308... 

7.04 

2.309760 

17.04 

-.058231... 

7.68 

.053275288 

24.72.. 

-.004955712 

.9441 

25.6641 

.9522 

ca.6163.. 

.021844 

23.037044 

.021348 

26.658992 

Remarks. — The  general  features  of  the  process,  being  the  same  as  heretofore 
given  {270,  Example,  need  no  further  explanation  than  they  have  already  re- 
ceived. Each  decimal  figure  of  the  root  is  added  the  first  time  in  the  first 
column  simply  by  annexing  it. 

In  finding  the  second  figure  of  the  root,  we  have  — ^,^  -  = tk"=  ^-G-     ^^^ 

this  cannot  be  the  proper  addition,  since  we  know  that  the  root  lies  betwefiii  4 
and  5  ;  hence  this  trial  fails  to  give  the  second  figure  in  the  root.  (See  273*) 
But  as  we  know  that  this  figure  cannot  be  greater  than  9,  we  try  9,  and  find 
that  it  makes  the  absolute  term  change  sign  so  that /(«)  and /'(«)  have  the 
same  sign,  and  consequently  .9  is  too  much  to  add.  (See  270fa.nd  also  consider 
that  f{x)  would  thus  be  shown  to  change  sign  as  x  passed  from  4  to  4.9,  and 
hence  that  a  root  lies  between  4  and  4.0,  244.)  We  therefore  try  .8,  and  find 
that  it  is  the  correct  addition.  We  know  that  .8  is  right,  since  we  know  that 
as  X  passes  from  4.8  to  4.9,  f{x)  changes  sign. 

In  finding  the  third  figure  we  have  for  trial  —^77— J  = '        =  .00.     Try- 

/  (a)  M.la 

ing  9  as  the  third  figure  of  the  root,  we  find  that  the  absolute  term  does  not 

change  sign,  and  hence  we  ktwio  that  9  is  the  next  figure,  i.  e.,  we  know  that  a 

root  lies  between  4.89  and  4.9. 

The  process  may  be  thus  continued  indefinitely,  and  as  many  figures  found  as 

we  may  desire. 

277»  N.  B. — It  will  be  observed  that  this  process  is  simply  one  of  substitu- 
tion in  f{x)  of  values  for  x  which  come  nearer  and  nearer  to  making  f{x)  —  0. 


HOIINERS   METHOD.  239 

Thus  in  this  example  4,  substituted  in  a- '  —  4a:*  —  6a;  +  8,  gives  x^  —  4x^  —  Qx 
+  8  =  —  16  ;  4.8  substituted  for  x,  gives  a;'*  —  4a;*  —  6a;  +  8  =  —  2.368 ;  4.89 
gives  «'  -  4x^  -  Qx+S=  -.058231 ;  4.892  gives  a;^-4a;2-6a-+8=  -.004955712. 
Thus  we  are  coming  nearer  and  nearer  to  the  number  which  substituted  for  x 
would  make  x^  —  4a;*  —  6a;  +  8  =  0,  or  would  satisfy  the  equation. 

2.  To  FIND  THE  ROOT  WHICH  LIES  BETWEEN  —1  AND  —2,  we  take  the  equa- 
tion x'^  +  4a;*  —  6a;  —  8  =  0  (changing  the  signs  of  the  terms  containing  the  even 
powers  of  x),  and  find  the  root  of  this  equation  which  lies  between  1  and  2 

(246\ 

OPERATION. 


h4 

-6 

-8                      1  1.8004H 

1 

5 

-1 

5 

-1 

-9... 

1 

6 

8.992 

6 

1 

5... 
6.24 

008 ••• 

.007249504064 

7.8 

11.24 

-.000750495936 

.8 

6.88 

8.6 

18.12 

.8 

.00376016 
18.12376016 

9.4004 

.0004 

.00376032 

9.4008 

18.12752048 

.0004 

9.4012 

o.  To  FIND  THE  ROOT  WHICH  LIES  BETWEEN  0  AND  1.  We  first  find  thc 
initial  figure  either  by  evaluating  f{x)  successively  for  .1,  .2,  .3,  etc.,  and  no- 
ticing when  it  changes  sign  (244) ;  or  by  Sturm's  method.  The  former  is  much 
the  less  laborious,  and  is  to  be  preferred  (2^S).  In  fact,  to  use  Sturm's  method 
involves  exactly  the  same  work  as  the  former  method,  w\i\\  considerable  additional 
work.  Moreover,  the  former  method  can  be  applied  mentally  till  the  proper 
initial  figure  is  determined,  and  no  other  writing  will  need  to  be  involved  than 
just  what  Horner's  method  requires.  No  figures  will  need  to  be  written  but 
those  in  the  following 

OPERATION. 


-4.. 

-6... 
-2.79 

+8...'                 1  .9082-f 

.9 

-7.911 

-3.1 

-8.79 

.089 

.9 

-1.98 

-086242688 

-2.2 

-10.77.... 

.002757313 

.9 

.010336 

-1.3.. 

-10.780336 

.003 

-1.292 

240  ADVANCKD   COUKSK   IN   ALGEBIIA. 

It  irt  so  evident  that  the  last  figure  is  2,  that  the  operation  for  verifying  it  i» 
unnecessary. 

2.  Find  the  roots  of  x'  —  13a;*  +  53.'c'  —  49a;'  —  110.?;  +  150  =  0, 
extending  the  decimals  to  the  5th  phice. 

Suo. — Apply  Sturm's  method.  If  there  arc  equal  roots,  depress  the  equa- 
tion. 

3  to  5.  Find  all  the  real  roots  of  the  following,  extending  the 
decimals  to  4  or  5  places: 

(3.)  7?  +  lOar*  +  bx  -  260  =  0 ; 

(4.)  ^4-    3a;* +  5^=  178;  \ 

(5.)  ^+    23;*  =  23a;  4-  TO. 

Tiie  cuhic  equations  on  pages  223,  224, 22G,  228,  will  afford  further 
exercise. 

G.  Find  the  roots  of  the  equation  a^  —  SOa;*  +  lOOSrc*  —  14937:*; 
+  5000  =  0. 

Sug's. — Oi  course  we  may  always  find  the  number  and  situation  of  the  real 
roots  by  Sturm's  method.  But  as  tlie  labor  of  t^ubstituting  in  aU  the  functions 
used  in  this  method  is  frequently  great,  we  avoid  it  when  we  can.  Howecer,  it 
is  generaUy  heM  to  free  tJie  equation  from  equal  roots,  and  find  the  number  of 
posUioe,  and  the  number  of  negaticc  roots  by  Sturm's  method.  But  the  situation 
of  the  roots  is  almost  mlways  more  readily  found  by  inspection  based  mainly  on 
the  change  in  sign  of  f{x)  {244).     We  will  solve  this  example  in  this  way. 

1 .  By  Sturm's  method  we  find  vhat  our  equation  has  no  equal  roots,  and  that 
it  has  4  positive  roots,  and  no  negative  root  (see  264). 

2.  We  now^  proceed  to  find  the  least  root.  Observing  that  for  x=0, 
f{x)  is  -h,  and  for  x=\,  f{x)  is  — ,  we  know  that  at  least  one  real  root  lies 
between  these  limits.     To  find  it  we  have  the  following  (see  next  page) : 


HORNER  S   METHOD. 


2^ 


FIRST   OPERATION. 


1 


-80 
.1 

+1998 
-   7.99 

-14937 
199.001 

+5000 
-1473.7999 

.1 

.2 

-79.9 
.1 

-79.8 
.1 

1990.01 

-  7.98 
1982.03 

-  7.97 

-14737.999 
198.203 

-14539.796* 
391.636 

3526.2001* 
-2829.6320 

696. 5681.... t 
-  274.4238,5424 

.3 
.02 

.02 
.01 

-79.7 
.1 

1974.06* 

-  15.88 

-14148.160 

388.468 

422.14424576} 
-  272.88640240 

.35 

-79.6* 
.2 

1958.18 
~  15.84 

-13759. 692.. .f 
38.499288 

149.25784336§ 
-  135.86783711 

-79.4 
.2 

1942.34 
-  15.80 

-13721.192712 
38.467784 

13.390006251 

-79.2 
.2 

1926.54-. t 
-   1.5756 

-13682. 724928t 

38.404808 

-79.0 
.2 

19249644 
-   1.5752 

-13644.320120 
38.373336 

-78.8-f 
.02 

1923.3892 
-   1.5748 

-13605.946784§ 
19.163073 

-78.78 
.02 

1921.8144t 
-   1.5740 

-13586.783711 
19.155211 

-78.76 
.02 

1920.240 1 
-   1.5736 

-13567.6285001 

-78.74 
.02 

1918.6668 
-   1.5732 

-78.72t 
.02 

1917.09361 
-    .7863 

-78.70 
.02 

-78.68 
.02 

1916.3073 

-  .7862 

1915.5211 

-  .7861 

-78.66 
.02 

1914.7350t 

-78.64§ 
.01 

-78.63 
.01 

-78.62 
.01 

• 

-78.61 

.01 

-78.60 

249  ADVANCED  COURSE  IN  ALGEBRA. 

Remarks.— This  work  is  given  to  show  how  we  may  proceed  to  find  the  first 
twj  figures  of  the  root  by  successive  simple  approximations.  If  the  student  is 
familiar  with  the  principles  heretofore  developed  and  applied,  he  will  have  no 
difficulty  ill  seeing  the  reasons  for  the  operations  above.  We  are  simply  adding 
to  the  value  of  x  substituted  in  /(r),  so  as  steadily  to  diminish  the  absolute 
term,  being  careful  not  to  add  so  great  an  amount  to  x  as  to  make  this  term 
change  its  sign;  and  when  we  can  add  no  more  of  one  order  (as  of  tenths),  we 
pass  to  the  next  lower  order  (hundreths)  and  proceed  in  the  same  manner.  On 
this  process  we  make  two  remarks,  viz. : 

(a.)  It  is  not  sure  to  succeed.  Thus,  if  there  were  two  roots  between  .34  and 
.35,  for  example,  the  absolute  term  would  not  change  sign  when  we  passed  from 
.34  to  .35,  although  we  would  have  passed  both  roots  ;  and  it  might  occur  that 
no  root  lay  beyoud  .35,  in  which  case  our  method  would  be  fruitless.  But  such 
cases  are  rare.  It  is  in  such  cases,  and  in  such  only,  that  Sturm's  method  is 
well-nigh  indispensable  for  finding  the  situation  of  roots. 

(6.)  In  most  cases  the  ex{tct  figure  of  any  order  can  be  told  without  such  an 

approximation  as  the  above  ;  or,  what  is  equivalent,  without  trying  a  figure,  and 

when  it  is  found  incorrect,  erasing  the  work  and  trying  another,  and  so  on  till 

the  right  figure  is  found.    In  this  particular  case,  the  first  figure  in  the  root  being 

a  small  fra/'tion,  the  higher  powers  of  x  might  be  neglected  (and  more  especially 

us  they  differ  in  signs),  and  —  14937j  +   5000  =  0  would  give  the  first  figuro 

5000 
In  the   root  at  once.     Thus  x  =  ;,  ,^.,-,  =.3  +.     So,  in  this  case,  for  the  second 

14937 

f  (a)  696.5681  "     ^^  i .  ,      .         .,  .  ^  ^  .,. 

figure Tn~~  — fj?t;Q  «Q»~-^  "•"'  ^^"icli  gives    the  next   figure  of  the 

J  Kfi)  — Io7o9.o9i* 

root. 

3.  To  FIND  THE  NEXT  GREATER  ROOT.  By  Substituting  l,we  find,  as  on  the 
next  page,/(ar)  =  —  8018 ;  and  when  1  is  added  to  this,  /(a*)  =  —17506.  Now  it 
is  evident  that  any  slight  addition,  as  of  2,  3,  or  4,  to  the  value  of  x,  will  only 
make  /(j;)  increase  negatively.  This  is  seen  by  inspecting  the  coefficients  1, 
—72,  +1542,  —7873,  —17506.  We  therefore  make  a  considerably  larger  addi- 
tion to  X,  as  10.  From  this  explanation  the  student  should  be  able  to  see  the 
significance  of  the  following  (see  next  page  :) 


HORNER  S   METHOD.  ^4S 


SECOND   OPERATION. 


-80 

+1998 

-14937 

+  5000 

1 

1 

-  79 

1919 

-13018 

1 

-79 

1919 

-13018 

-  8018 

10 

1 

-  78 

1841 

-  9488 

12.7 

-78 

1841 

-11177 

-17503 

1 

.-  77 

1689 

1;M70 

1764 

-  403G  .... 

-77 

-  9488 

1 

-  75 

1615 

3737.3441 

-76 

1689 

-  7873 

-  298.655d 

1 

-  74 

9220 

-75 

1615 

1347 

1 

-  73 

4020 

-74 

1542 

5367... 

1 

-  620 

-   27.937 

-73 

922 

5339.063 

1 

-  520 

-   42.931 

-72 

402 

5296.133 

10 

-  420 

-62 

-  18.. 

10 

-  21.91 

^52 

-  39.91 

10 

-  21.42 

^42 

-  61.33 

10 

-  20.93 

-32. 

-  82.20 

-31.3 

.7 

-30.6 

.7 

^d~A) 

.7 

As  now  f{a)  and  /  '{a)  have  opposite  signs,  and  tlie  remainder  of  the  root  is 
quite  small  as  compared  with  that  already  found,  the  approximation  can  be 

.,.,,.                      m,              ,              /(«)           -298.6559      ^, 
earned  on  m  the  ordmary  way.     Thus  we  have  —         •  = ^-   —.05+, 

and  the  next  figure  of  the  root  is  5. 

4.  To  FIND  THE  NEXT  GREATER  ROOT  we  resume  the  coeiRcients  after  the 
roots  had  bee  i  diminished  by  12.     Then  adding  1  to  the  value  of  cc,  we  find  that 


244 


ADVANCED  COURSE  IN  ALGEBRA. 


for  .c  =■  13,  /(ar)  =  1282,  having  changed  sign,  as  it  should.  Now  as  f\x),  i.  e. 
52^9,  aud/(jr)  are  both  positive,  and  the  other  coeflBcients,  though  negative,  are 
comparatively  small,  it  will  take  considerable  increase  in  x  to  change  the  sign 
of  f{x).  We  therefore  add  10.  Now  f\x)  has  changed  sign,  and  by  inspecting 
the  coefficients,  1,  +12,  —348,  —1321,  and  24872,  it  is  evident  that  x  cannot  in- 
crease another  10  without  changing  the  sign  of  f{x).  Hence  we  try  5.  For 
a  similar  reason  we  add  4  next. 


TIIIUD    OPERATION. 


-82 

-   18 

+5367 

-  4036 

12 

1 

-  31 

-  49 

5318 

1 

-31 

-  49 

5318 

1282* 

10 

1 

-  30 

-  79 

23590 

5 

-30 

-  79 

-  29 

5239* 

-2880 

24872t 
-13180 

4 

1 

32.+ 

-29 

-108* 

2359 

11692t 

1 

-180 

-3680 

-11588 

-28* 

-288 

-13211 

104§ 

10 

-80 

-1315 

-18 

-368 

-2636 

10 

20 

-  765 

-  8 

-348t 

-3401t 

10 

85 

504 

2 

-263 

-2897 

10 

110 

1144 

12t 

-153 

-1753§ 

5 

135 

17 

-  18t 

5 

144 

; 

22 

126 

5 

160 

27 

286 

5 

176 

m 

462§ 

4 

86 

4 

40 

4 

44 

.^ 

4 

4^ 

f'i^') 

But  as  the 

coefficients  preceding  — 

1753  ar 

hornek's  method.  246 

all  4-,  they  will  diminish  it  somewhat  in  the  operation,  and  hence  it  is  probable 
that  .06  is  the  proper  addition  to  make  to  the  root.  The  process  can  now  be 
continued  to  any  extent  desired. 

5.  To  FIND  THE  NEXT  GREATEST  (in  this  casc  the  greatest)  root,  we  have  the 
following  operation,  which  we  leave  the  student  to  trace: 

fourth  operation. 

32 


^-48 

+462 

1 

49 

49 

511 

1 

50 

50 

561 

1 

51 

51 

612* 

1 

53 

52* 

665 

1 

54 

53 

719 

1 

55 

54 

774. . . 

1 

45.44 

55 

819.44 

1 

46.08 

56.8 

865.52 

.8 

46.72 

57.6 

912.24 

.8 

58.4 

.8 

-1753 
511 

+  104 
-1242 
-1138* 
16 

1 
1 

-1242 
561 

34.8  + 

-  681* 
665 

-1154..... 

1086.8416 

-  67.1584 

-  16 
719 

703.... 
655.552 

1358.552 
092.416 

2050.968 


59.2 

The  student  should  extend  these  solutions  2  or  8  figures  farther. 

7  to  12.  Solve  the  following: 

(7.)  ^  +  ^^x"  -  800a;  =  60000. 

(8.)  ar'  +  2a;^  +  3a;^  +  4:^:^  +  5a;  =  64321. 

(9.)  a;*  4-  4a;'  -  4a:*  -  llrr  +  4  =  0. 
(10.)  a^  _  27a:'  +  ir)2a:«  +  356a:  =  1200. 
(11.)  x"  -  Zx'  =  48654231721. 
(12.)  .a:'  +  2a:'  +  3a:  =  13089030. 
(13.)  of  -  10.r^  -f  6a:  =  1. 
(14.)  .^•'  +  173a:  =  14760038046. 


246  ADVANCED  COURSE  IN  ALGEBRA. 

(15.)  re*  -  7035a:*  +  15262754^  =  10000730880. 

(16.)  x'  -H  Ux  =  35.4025.    (Solve  by  Horner's  method.) 

(17.)  ic^  +  4a;«  -  9a;  =  57.623625. 

(18.)  2a;*  +  ba^-{-  4a:' 4-  3a;  =  8002.  (Observe  that  it  is  not  neces- 
sary to  make  the  first  coefficient  unity.  See  examples  in  the  firsu 
part  of  the  section.) 

(19.)  3a;*  -  4a;*  +  2a;  =  1000. 

(20.)  5a;^  -  3.2a;  ==  41278.216. 

Note. — The  roots  of  several  of  the  above  are  commensurable ;  and  their  solu- 
tion shows  that  Horner's  method  is  adapted  to  such  caseg. 


21  to  25.  Extract  the  roots  of  the  following  numbers  by  Horner's 
metliod : 

(21.)  The  cube  root  of  119736852154. 

(22.)  The  square  root  of  5126485. 

(23.)  The  fifth  root  of  2. 

(24.)  The  fourth  root  of  35718271002567691. 

(25.)  The  cube  root  of  3. 

Sug'8. — To  solve  the  21st,  write  a;'  —  119736852154,  and  solve  as  usual,  being 
careful  to  remember  that  the  coefficients  are  1,0,0,  —119736852154.  To  find 
the  initial  figure,  point  off  as  in  the  ordinary  method  of  extracting  roots.  The 
following  exhibits  the  first  steps  of  the  process : 

[  49 


0 

0 

-11973C852154 

4 

16 

64 

4 

16 

-  5573G 

4 

82 

53649 

8 

48- 

-  2087852 

4 

1161 

129 

5961 

9 

1242 

138 

7208 

9 

147 

26  to  29.  Solve  the  following  by  first  eliminating,  and  then  solving 
the  resulting  equation  by  Horner's  method: 

(26.)  2.7;'  —  5a;  +  3y  =  2a-//  —  4a;*  +  12,  and  4?/'  —  3a;  =  2y  +  5. 
(27.)  '2if  -  \xy  +  2a;'  -  3//  -  2a;-8  ^-  0,  and  4/  +  4a;-^  =r  11. 
(28.)  If  -  ^xy  +  2a;'  -  3//  -  2a;  =8,andi/2  x  2«/  +  a;2— 6a;=  —  6. 
(29.)  2/  -  ^xy  +  2a;'— 3y  -.2.c  =  8,  and  ?/2^6y  +  .c'-4a;  +  9  =  0. 


HORNER  S    RULE. 


247 


Sug's.— From  the  2d  of  (26)  we  have  y  =  \±  ^ySx  +  V-     Substituting  this 
in  the  1st,  we  obtain  Qx*  —  \^x  -  a/.  =  {x-i)^dx  +  \^,  whence    36a;*   -  69a;' 


-  101a;2  +  ^^x  +  H^=0.     And  dividing  by  36,  we  have  x*  - 
4-  3,687a;  -|-  3.188  =  0,  carrying  the  fractions  to  three  places. 


1.917a;='  -  2.806a;« 


27 S,  ScH. — There  are  various  methods  by  which  Honier's  process  may 
be  abridged,  especially  when  a  large  number  of  decimals  is  required ;  but 
we  have  thought  it  better  to  exhibit  fully  and  clearly  the  principles  essentia) 
to  the  process,  than  to  spend  time  and  distract  attention  by  giving  these 
arithmetical  abridgments.  The  most  simple  of  these  are :  {a)  the  omission 
of  the  decimal  point ;  (h)  the  writing  of  the  sums  only  in  the  several  working 
columns,  performing  the  various  multiplications  and  additions  mentally; 
(c)  after  several  decimals  have  been  obtained,  instead  of  annexing  0'  ^ 
(or  •••  's)  to  the  working  columns,  d?'opping  off  ^giires  from  the  right  iw 
each  new  operation,  as  one  from  next  to  the  last  right-hand  column,  two  from 
the  next  to  the  left,  three  from  the  next  to  the  left,  etc.  ;  {d)  and,  tinally, 
when  all  the  working  columns  but  the  last  two  have  disappeared,  continu- 
ing the  operation  as  a  process  of  simple  division,  only  dropping  off  a  figure 
from  the  right  of  the  divisor  at  each  step  instead  of  annexing  a  0  to  the 
dividend.     We  condense  an  example  from  Todhunter  as  an  illustration. 


Ex. — To  compute  to  16  decimal  places  the  root  ot  a?  +  dx^ 
—  5  =  0,  which  lies  between  1  and  2. 


2x 


OPERATION. 

3 

-3 

-5                        j  1.3300587305679821 

4 

2 

-3000 

5 

700 

-333000 

CO 

889 

-663000000000 

63 

108700 

-98647524875 

66 

110779 

-8347885443 

690 

112867000000 

-446624425 

693 

112870495025 

-107998801 

696 

112873990075 

-6411112 

699000 

11287454929 

-767351 

699005 

11287510850 

-90100 

699010 

1128751574 

-11087 

699015 

1128752063 

-929 

112875208 

-27 

112875210 

-4 

218  ADVANCED   COUllSE   IN   ALQEBliA, 


SECTION  III. 
GENERAL  SOLUTION  OF  CUBIC  AND  BIQUADRATIC  EQUATIONS. 


CaKDAN'S   SOLUTIOif   OF   CuBIC  EQUATIONS. 

;S?7.9,  Prob* — To  resolve  the  general  cubic  equation  x'  +px' 
+  qx  +  r  =  0. 

Solution. — This  solution  consists  of  three  steps:  1.  To  transform  the  equa- 
tion into  one  of  the  form  y*  -f-  my  -\-  n  =  Q,  tliat  is,  an  incomplete  cubic  lacking 
the  square  of  the  unknown  quantity.  To  effect  this,  we  put  x  =  y  -\-  z,  and 
substituting,  have 

y^  +  dy^z  +  Syz^  +  2'  +  py^  +  2pyz  +  pz*  -\-  qy  +'qz  +  r  =  0, 
ar,y'^ -\-{3z+p)y* +  (^z* +2pz+q)y  +  z^ -j-pz"^ +qz  +  r  =  0.  (1) 

Now  as  we  have  only  one  condition  expressed  between  y  and  z,  viz.,  y+z—x, 
we  are  at  liberty  to  impose  another.  Let  us  put  82  + 1>  =  0,  whence  z  =  —  ^p. 
Tlien  will  this  value  of  z  substituted  in  (1)  give 

y'  +  (7  -  ip')y  +  (rrP'  -ipq  +  r)  =  0.  (2) 

2.  Since  the  above  transformation  can  always  be  effected,  a  solution  of 

y""  +  my +  11=0  (3) 

will  include  the  solution  of  all  cubic  equations.  Our  second  step  is  to  trans- 
form this  equation  into  one  which  can  be  solved  as  a  quadratic.  To  do  this  we 
put  y  =  u  +  V,  which  gives  (3)  the  form 

u*  4-  Su*v  -f  3mo'  -f-  v^  -4-  m{u  -+- 1;)  +  72,  =  0, 
or,    u*  +  Sut{u  +  t)  +  v'-^  -h  v\{u  +  t?)  4-  71  =  0, 
or,    u*  -k-v*  -\-  (3mp  -h  mXu  +  t)  +  n  =  0.  (4) 

Now,  as  we  have  but  one  condition  expressed  between  u  and  v,  viz.,  u+v=y, 
we  are  at  liberty  to  impose  another.     Let  us  put  Zuv  +  m  =  (i,  whence  v  = 

—  — ;  and  (4)  becomes  u^  ->t- v^  +  n  =z  0, 

ou 

or  by  substituting  the  value  of  t, 

-        m^ 

»    -27^  +  ''  =  "*' 

whence  we  have  u^  -\-  nu*  =  /rm\  (5) 

3.  Solving  this  quadratic  we  obtain 


and  as  tj'  =  —  {\l^  4-  w),     v  =  4/—^^  T  -v/jV^*^  +  4^*. 


CARDAk's   SOLUTION'  OF   CUBIC   EQUATIONS.  249 

Finally,  taking  the  square  root  as  +  for  the  value  of  u,  and  -  for  the  value 
of  V,  since  these  are  to -responding  values,  we  have 


y  =  1^  -  i^  +  V-^7^'  +  in^  +  ^  ~\n-  A^-hm^  +  \nK 


(6) 


280.  Vrop. — 1.  In  the  equation  y^  +  my  +  n  =  0,  when  m  ii 
positive,  and  when  m  is  negative  and  ^m^  <  ^^n'^  the  equation  has 
one  real  and  two  imaginary  roots,  and  GardarCs  formula  (6)  gives 
a  satisfactory  solutloti, 

2.  When  m  is  negative  and  ^m^  =  Jn',  tico  of  the  roots  are  equal, 
and  Cardan^s  method  is  satisfactory.* 

3.  But,  when  m  is  negative  and  ^m^  >  Jn^  all  the  roots  are  real 
and  unequal,  while  Carda?i*s  method  makes  them  apparently  imagi- 
nary, and  the  solution,  is  unsatisfactory. 

Dem. — A  cubic  equation  must  have  at  least  one  real  root  {238).  Let  this  be 
a.  Now  conceive  the  equation  reduced  to  a  quadratic  by  dividing /(.i)  by  x—a, 
and  let  6  +  /y/  c,  and  b  —  ^^  c  be  the  roots  of  this  quadratic,  these  being  the 
general  forms  of  the  roots  of  a  quadratic,  in  which  if  c  is  +  the  roots  are  real, 
if  c  is  —  they  are  imaginary,  and  if  c  is  0  these  two  roots  are  equal. 

Now,  a,  b  +  ^Z  c,  and  b  —  \/  c  being  the  roots  of  the  equation,  we  have 
by  (235) 

(x-a)  (x-[b+'^c])  {x-[b-^c])=x^-{a+2b)x^+{2ab+b^-c)x-a{b^-c)=0. 

To  transform  this  into  the  form  y^  +  my  +  n  =  0,  we  must  put  a  +  2b  =  0; 
whence  a  =  —2b,  and  we  have 

y-«  -  (3^2  +  c)y  +  2b{b^  -  c)  =  0. 

Comparing  this  with  Cardan's  formula,  we  see  that 


Hence  we  see  that  if  c  is  +,  that  is,  if  all  the  roots  of  a  cubic  etiuation  aro 
real  and  unequal,  Cardan's  method  gives  a  result  apparently  imaginary.  But  if  c  is 
— ,  that  is,  if  two  of  the  roots  are  imaginary,  Cardan's  method  gives  s^real  form. 
Also  when  c  =  0,  that  is,  when  the  roots  are  a,  6,  and  6,  the  form  is  real,  since 

Now  by  inspecting  the  quantity  -y/ -.lim'  +  W  we  see  that  it  is  real  when 
m  is  positive  ;  and  also  when  m  is  negative  if  ^m^  <  {n"^.     Hence  in  these 


»  If  flW  the  roots  aro  equal,  the  equation  talcos  the  form  {x  -  a)3  =x^  -  3a.r;2  ^  Za"'X  -  a^  =  0, 
a  being  the  value  of  one  of  the  equal  iO'>tB  C-iSn).  In  Miis  case  the  transformation  which  makes 
the  term  m  z"^  disappear  gives  y'  =  0,  since  a;  =  y  -  ip  =  y  +  a,  and  y  =  x  -  a  =  0. 


25©  Anxk^cKV  coyr^SE  in  ^lqecra. 

cases  there  are  one  real  &ix(\  two  imaginary  roots,  and  Cardan's  inethod,-giving 
a  real  form,  enables  us  to  determine  one  of  them,  and  hence  to  solve  the 
equation. 

2d.  We  have  also  seen  above  that  when  c  =  0,  that  is,  when  two  of  the  roots 


are  equal  (and  not  all  three),  'y/Aw*  +  \n*  =  0,  in  which  case  m  must  be  nega- 
tive and  -^rm^  =  in*. 

3d  It  has  also  appeared  above  that  when  all  the  roots  are  recU  and  unequal,  Car- 
dan's method  gives  an  apparently  imaginary  result.  But  this  can  only  be  the  case 
when  rfi  is  negative,  and  -^Sm^  >  \n*. 

28 1,  ScH. — Cardan's  method  would  seem  to  give  a  cubic  equation  nine 
roots  instead  of  three,  since  as  there  are  three  cube  roots  of  any  number, 

a/  —  in+  \/-A»»"*  +  i"-*  would  liave  three  values,  and  4/-7W-  -y/ A"*^  +  \n* 

would  have  three  other  values.  Now  combining  each  of  the  former,  in 
turn,  with  each  of  the  latter,  we  should  have  vitte  results.  In  order  to  ex- 
plain this  seeming  paradox,  let  us  find  the  form  of  the  three  cube  roots  of  a 
number,  as  of  a'.  To  do  this  we  have  but  to  solve  the  equation  x'^  =a^. 
Thus  x^  —a^  =  {x  —  a){x*+ax-{-  a*)  =  0.  Whence  x  —  a  =  0,  and  x'+ax+a* 
=0.  From  these  we  have  x  =  a,  —^aCi  +  's/— 3),  and  —  ^a (1  — /y/ —  8). 
Now  let  the  roots  of  a/  ^\n-\-  ^J^-m*  +  \n^  be  r,  —  ir(l-f  \/— 3),  and 

—  ir(l-V^^);    and   the   roots   of  k/  -\n  -  V^Vw*  +  \n^  be  r' ,  -\r' 

(1  +  -y/  — 3),  and  —  \r'  (1  —  y'--3).    It  will  be  remembered  that  we  assumed 

««  =  —  —;  that  is,  the  products  of    the    admissible    roots    must    be    real. 
o 

Therefore  we  can  use  for  the  parts  of  the  root  r  and  ?•',  —  \r{\  +  >y/  —  8)  and 

—  \t'  iX  —  'v/  — 3),  and   —  \r{\  —  ^  —  \)  and   —  ir'(l  4-  -y/  —  8);  and  we  can 

use  these  parts  in  no  other  combination,  as  any  other  would  not  give  a  real 

quantity.      Thus  we  cannot  have  y  —n  -k-  t  —  r  -\r  {\  4-  y'  —  3),   since    wo 

would  then  be  —  r[|r(l  -f-  y^ — 3)],  which  is  an  imaginary  quantity,  and  hence 

fix 

not  equal  to  —  — ,  as  it  should  be. 

o 

We  will  give  a  few  examples  to  which  the  student  may  apply  Cardan's  pro 


Examples. 

Solve  the  following,  finding  one  of  the  roots  by  Cardan's  process, 
and  then  depressing  the  equation  by  division,  solve  the  resulting 
quadratic. 


DESCARTES* S   SOLUTION   OF  BIQUADRATICS.  25X 

1.  a^  -  9x  +  28  =  0.* 

2.  a:f^  —  3x^  +  4:  =  0.     (See  first  step  in  general  solution.) 

3.  a^  —  Gx  +  4:  =  0. 
4:,  x"  +  6x~2  =  0. 

6.  X  +  b  +  3  \/^  =  a. 
Q.  x"  +  3x'  +  dx  -  13  =  0. 
Z  a^-9x'  +  6x-2  =  0. 

8.  x'  —  6x'  +  13a;  -  10  =  0. 

9.  x^  -  48a;  =  128. 

10.  x'  +  2z  =  12. 

11.  z'  -3:^-  2z'  -  8  =  O.f 


12. 
13. 

2:^ 

7?  - 

-  Qy'  +  13?/  = 

-  12a:^  +  36a: 

12. 
=  44. 

11 

1  +  X       ^  a  +  X 

^/x 

15. 

a        '^         X 
-  '^x'  +  \^X  ~ 

12  = 

c 

:  0. 

SuG. — An  attempt  to  solve  the  last  by  Cardan's  process  will  give  roots 
apparently  imaginary,  although  it  is  easy  to  see  that  the  roots  are  all  real,  and 
commensurable. 


Descartes's  Solutioj^  of  Biquadratics. 

282,  Prob. —  To  resolve  the  general  biquadratic  equation  x*  + 

ax'  +  bx''  +  dx  +  e  =  0. 

Solution. — The  first  step  in  the  process  is  to  transform  the  equation  into  one 
wanting  the  3d  power  of  the  unknown  quantity.  This  is  done  in  the  usual  way 
(see  Cardan's  method  of  resolving  cubics) ;  i.  e.,  by  putting  x=y  -{-z,  substituting, 
collecting  the  coefficients  with  reference  to  y,  and,  putting  the  coefficient  of  y ' 
equal  to  0,  finding  the  value  of  z.  This  value  of  z  substituted  in  the  given 
equation  will  give  the  form 

y*  +  my"^  +  iiy  +  r  =  0. 
2.  Assume  y*  +  my^  +  ny  +  r  =  (y^  +  cy  +  /)  (y^  +  ey  +  g),    and    deter- 


*  It  is  bttlcr  for  the  ptudent  to  use  Cardan's  processXhan  to  substitute  in  the  formula.    Tliii« 

for  JC»  -  Ox  +  28  =  0,    we  have,  by  putting  a  =  y  +  s,  y'  +  8^  +  (%2  -  9)  (2  +  y)  +  2S  -.r  o ; 

3  27  3 

and  making;  Zyz  -  9  =  0,  or  2  =-,  i/S  +  -?  +  28=0.    Whence  y=  -1,  and  -3,  and  2  =  -  =  -3,  and 
y  y*  y 

-  1.    .'.  X  =  y  +  2  =  -  4.    Then  {x^  -  9x  +  28)  ■+■  (a:f  4) .-  a;^  -  4a;  +  7  =  0 ;  whence  a-=\!±  |/^. 

t  An  equation  of  the  form  x'^"^  +  ax'i"^ -\  bx'»  +c  =  0  can  be  reduced  to  a  cubic  of  the  form 
y»+my+n=0,  by  putting  x»t=y- ^<z. 


252  ADVANCED   COURSE   IN   ALGEBIIA. 

mine  the  quantities  r,  c,f,  and  g,  ^o  that  they  will  fulfill  the  required  conditions 
Thu."^,  expanding  we  have 

y^  +  my^  +  ny  +  r  =  y*+  c     y^  +  f     y^  +  ef    y  +  fg  ; 


+  c 

y'+f 

r+ef 

+  e 

+  CC 

+  /7 

+  rg 

whence,  as  the  members  are  identical, 

c  +  e  =  0,        f  +  ce  +  g  =  m,        ef  +  eg  =  n,        and    fg  =  r. 
From  the  first  we  see  that  c  =  —  e.     Substituting  this  value,  we  have 

(1)    f-e*  +  g  =  m;        (2)    e(f-g)=n;        and  (3)    fg  =  r. 

From  (1)  and  (2)  wt-  have  g  =  Ue^  —  ~  +   m\  and  /=  i^e'  +  -  -f-  mj; 

which  substituted  in  (3)  give      (e^  +  -  +  m)    (e^  _  ?-  +  mj    =  c*  +  2me" 

n* 
;  +  wi*  =  4?-,   or 

6* 

e^  +  2me*  +  {in'  -  Ar)e^  -  n^  =  0.  (4) 

Now  (4)  can  be  reduced  to  a  cubic  in  terms  of  c,  by  putting  «'=  ^,  —  Jm  (see 
foot-note  on  preceding  page).  Tliis  cubic  equation  will  have  at  least  one  real 
mot  {2'i8),  and  this  will  give  real  values  to  e,,and  hence  to  «,  r,/,  and  g. 
^VlK■refo^e,  */  Cardan'8  method  gives  a  practical  solution  of  (4),  we  can  resolve  the 
biquadratic. 

283,  Scir. — It  will  be  observed  that  this  resolution  of  a  biquadratic  in- 
volves the  resolution  of  a  cubic,  and  hence  is  subject  to  tlie  difficulty  attend- 
ing the  irreducible  case  of  cubics.  We  will  give  a  sin'_,d'.;  example,  to  wliich 
the  student  can  apply  the  process  of  Descartes. 

Ex.— Find  by  Descartes's  method  the  roots  of  x'—  lOx"—  20a;  —  16 
=  0. 


Recurring  Equations. 

284:»  A  jRecurring  Equation  is  an  equation  such  tliat  the 
coefficients  equidistant  from  the  first  and  last  are  numerically  equal, 
when  the  equation  is  in  the  complete  form  JaT-f  Bx''~'^-\-  Cx""'^ .  - .  - 
iy  =  0 ;  and  the  signs  of  the  corresponding  terms  are  either  all  alike, 
or  all  unlike ;  i,  c,  the  coefficients  of  the  first  half  recur  in  an  inverse 
order  in  the  second  half  of  the  function. 

III.  12a;'  -I-  Zx*^  —  hx^  —  5^*  +  3j;  +  12  =  0  is  a  recurring  equation. 
Ax^  +  Bx^-^  +  0"-5  ....  Cx'  +  Pj;  -h  ^  —  0  is  the  type  of  such  equations. 

28S.  Prop,  1, —  The  roots  of  (i  recurring  equation  are  recipro- 

cols  of  each  other ;  i.  e.,  ^/'    a   is   a   rooty  -   is  also ^  and  so  of  en ch 

a 

of  the  roots. 


RECURRING  EQUATIONS.  2^53 

Dem. — If  a  satisfiea  the  equation 

Ax"  +  i?ar»-»  +  Oc"-'  ...  -  Cx^  +  Bx  +  A  =  0, 
~  will  also  satisfy  it,  for  the  former  when  substituted  gives 

Aa^  4- \Sa~-i  +  (7a— «  -  -  •  -  Ca^  ^  Ba  +  Az^d-, 
and  the  latter  gives 

A  B     ^      G  OB, 

which,  by  multiplying  by  «»  becomes 

A  +  Ba  +  Ca^  .  -   -  -  Ca""-^  +  ^a"-'  +  ^a«  =  0, 
a  result  identical  with  that  obtained  when  a  is  substituted. 

280,  ScH. — From  this  relation  among  the  roots  of  recurring  equations, 
they  are  often  called  Recijwocal  Equations. 

287,  Cor.  1. — If  the  degree  of  the  equation  is  odd  the  correspond- 
ing coefficients  may  all  have  like,  or  all  unlike,  signs ;  hut,  if  the 
degree  is  EVEX  they  rnttst  have  like  signs  sinless  the  middle  term  is 
wanting,  in  which  case  they  may  have  unlike  sig7is,  and  the  roots 
still  be  recijyrocal. 

That  the  signs  may  be  unlike  in  the  cases  specified  is  evident  since,  if  in  such 

cases  a  is  a  root,  and  we  substitute  -  instead  of  a,  clear  of  fractions,  and  change 

all  the  signs,  we  shall  have  the  same  result  as  if  a  had  been  substituted.     Thus, 

if  substituting  rt  gives  ^a"  +  5a'*—C'a^  +  (7(r«-—J?«—yI=0,  substituting  -    will 

.     A      B       C      C     B       ,      ^       ,  ,      .       ,^.  ,  /*     . 

give  -r+  — : 3  +  — 7 A  =  (i:  whence  clearing  of  fractions  and  changing 

all  the  figns  we  have  —  A  —  Ba  +  Ca-  —  Ca^  +  Ba*  +  Aa^  =  0,  a  result  iden- 
tical with  the  former.  The  fact  concerning  the  equation  of  an  even  degree  ia 
shown  in  a  similar  manner.  Notice  that  all  the  corresponding  coeiiicients  must 
have  like  signs  or  nil  unlike  signs. 

2S8,  Cor.  2. — A  recurring  equation  may  always  he  reduced  to  a 
form  having  the  coefficient  of  the  highest  poioer  of  the  unknoicn 
quantity,  and  the  absolute  term,  each  1,  since  by  definition  these  are 
nnmerically  equal. 


28f)»  Prop,  2. — A  recurring  equation  of  an  odd  degree  has  one 
of  its  roots  —1  if  the  signs  of  the  corresponding  terms  are  alike,  and 
+  1  if  they  are  unlike. 

Dem.— Having  a;«±yl.c«-i  ±  Bx''-^±  Gc"-' .  .  .  .  ±  Cx'^  ±  Bx-  ±^^±1=0,* 

*  The  ?ij?n  of  x*  can  always  be  made  +.  The  ambiguous  signs  are  to  bo  taken  ■♦■  or  — ,  aO" 
cording  to  the  hypo'heiia. 


2^  ADVANCED  COURSE  IN  ALGEBRA. 

taking  the  signs  of  the  corresponding  terms  alike  we  can  write 

(a^  +  1)  ±  Ax{x*-^  +  1)  ±  Bx^x"-^  +  1)  ±  Cx'^ix*-^  +  1)  +  etc.  =  0, 
which  is  divisible  by  a;  +  1  (Part  I.,  119),  wherefore  —  1  is  a  root  {231). 
Taking  the  signs  of  the  corresponding  terms  unlike,  we  can  write 

(if  -  1)  ±  Ax(x^-^  -  1)  ±  Bx'ix'^-*  -  1)  ±  Qc^ix'^-^  -  1)  +  etc.  =  0. 
which  is  divisible  by  a;  —  1  (Part  I.,  119),  wherefore  +  1  is  a  root  {231). 


290,  I*r02^»  3, — A  recurring  equation  of  an  even  degree^  xohose 
corresponding  terms  have  opposite  signs,  has  o?ie  root  + 1,  a?id  one 
root  —1. 

Dem.— Having  a'"  ±  Ax^"-^  ±  J5a**"-'  ±  Cx^*-^  .  .  .  .  zf  Cx'3  T  Bx*  T  Ax 
—  1=0,  taking  the  signs  of  the  corresponding  terms  unlike,  and  remembering 
that  the  middle  term,  which  would  have  no  corresponding  term,  is  wanting 
{287),  we  can  write 

(a:^"  -  1)  ±  Ax  (a-'— » -  1)  ±  jBa;«(^"— '  -  1)  ±  Cx\x^''-^  -  1)  +  etc.  =  0, 
which   is  divisible   by  a:*  —  1   (Part   I.,   119);    wherefore    a;*  —  1  =  0,   and 
a-  =  +  1   and  —  1. 


29 1,  ^rop,  4. — A  recurring  equation  of  an  even  degree  above 
the  serondy  mag  be  reduced  to  an  equation  of  half  that  degree,  when 
the  signs  of  the  corresponding  terms  are  alike. 

Dem.— Having  a^" J: ^a-»—'±i?a^"-«± Ca:^—5  ....  ^j^^*  .  .  .  ±Ca?±Bx^±Ax 
+  1=0,  taking  the  signs  of  the  corresponding  terms  alike,  we  can  write 

(a-*"  +  1)  ±  ^(a;»— '  +  x)  ±  B{x**-^  +  x*)  ±  C{x^*-^  +  x^)  +  etc.  =  0 ; 
whence,  dividing  by  a?*,  we  have 

-   -   -   -   l(x  +  ^  ±M=0. 

Now  putting    a;  +    -   =  y,  we  can  write  Ix  +   -)    =  a;'  +  2   4-  -j  =  y', 

1  /1\^  111  1 

whence    x*  -\ r  =  y'—  2.      [x  +  -)  =a;'+3a;'-   +  dx  -^  +  —..   =  x^  -\ 

x*  \        x/  XX*      a;"*  x* 

+  six  +-)  =  y\    whence    a;'  +  — ^  =  y'  —  3y. 

(x*  -I-  ^2)'=  «*  +  2   +  -  =  (y«  -  2)S  whence  x'  +  ^^  =  {y^  -  2)»  -  2. 

( a;  +    - )    =  a;«  +  5.c*  -    +   lOa;-*  -,     +    10a;*  -.     +    Sa-  —    +    -    =  a;*  +    -. 
\  x/  XX*  a;*  a;*  a;"  a;^ 

+  5  (a;^  +  -J  j  +  10  I  X  +    -  j  =  y',  whence  x^  +  -^  =  y'  —  5(y'  —  3y)  —  lOy. 


BINOMIAL   EQUATIONS. — ROOTS   OF   UNITY.  25^ 

(^'  ^  ^0    =  ^'  +   ^   +  ^6  =  (y^-  W,   whence  ^'  +  ^,  =  {y'-  33^)^-2. 
Whence  we  see  that  any  term  of  the  fonn  a^+  —  may  be  expreesed  in  terms 

of  p,  and  will  involve  no  higher  power  than  y".  Therefore  the  original  equa- 
tion, which  is  of  the  2?2th  degree,  can  by  this  substitution  be  transformed  into 
an  equation  in  y,  of  the  7ith  degree. 

Examples. 

Solve  the  following  recurring  equations  by  applying  the  foregoing 
principles : 

1.  a^-5af+  6x'  —  5x  +1  =  0. 

2.  ar^  -  lla;^  +  ITr'  +  17x'  -  ll:c  +  1  =  0. 

3.  6x'  -  11a;*  -  S^x"  +  33a;'  +  11a:  -  6  =  0. 

4.  1  +x^=a(l  +xY, 

5.  x'  -2x^  -^  x'  +  x"-  2x'  +1  =  0. 

6.  Sx^  -  16.7.^  -  1h7?  -  I62:'  +8  =  0. 

7.  ^x'  -  24:0^  +  57a;*  -  732;'  +  57a;'  -  24a;  +  4  =  0. 

8.  X*  +  4«a.''  -  19aV  +  4«'a;  +  a*  =  0. 

9.  ar*  +  a;3  +  a;'  +  a;  +  1  =  0. 
10.  1  +  a;*  =  i(l  +  xy. 

Binomial  Equations  and  the  Roots  of  Unity. 

292.  A  Binomial  Equation  is  one  of  the  form  x"  d=  «  =  o. 
Such  equations  may  be  considered  as  recurring  equations  and  solved 
accordingly. 

III. — Having  ic*  ±  a  =  0,  put  x^  =  ay" ;  whence  ay^  ±  a  =  0,  or  y"  ±1=0, 
which  is  recurring. 

Examples. 


1.  ar'  =t  5  =  0. 

3.  a;»  ±  2  =  0 

5.  a;''±  11  =  0. 

2.  a;*  d=  3  =  0. 

4.  a;«  ±  7  =  0. 

e.  x'  ±     1  =  0. 

7.  What  are  the  two  square  roots  of  1?  The  three  cube  roots  of 
1  ?  The  four  fourth-roots  of  1  ?  The  five  fifth-roots  of  1  ?  The 
six  sixth-roots  of  1  ? 

SuG. — The  solution  of  these  questions  consists  in  resolving  a!^  —  1  =  0, 
j;3  —  1  =  0,    a;*  —  1  =  0,  etc.     The  five  fifth-roots  of  1  are 

1,  ^(i^5-l  ±V- 10-2  1^5),  and -i(t'5  +  l  ±V-10 +2  V5). 

293,  Sen. — It  will  be  observed  that  the  form  x^  ±1=0  is  omitted 
above.     Now  x^  —  1  =  0  has  one  root  1.     The  equation  can  therefore   Ixj 


256  ADVANCED  COURSR  IN  ALGEBRA. 

depressed  to  a  recurring  equation  of  the  6th  degree,  having  all  its  signs  + . 
This  can  be  reduced  to  a  cubic  by  {291).  a;'  +  1  =  0  has  one  root  rf-  =  —  1, 
and  can  be  reduced  to  a  recurring  equation  of  the  Cth  degree  having  its 
signs  alternately  +  and  — .  This  can  be  resolved  into  one  of  theS'rd  degree 
by  {291).  Hence  the  complete  resolution  of  a:^  ±  1  =  0  depends  on  the 
resolution  of  a  cubic. 

^9  ±  a  =  0  can  be  resolved  by  putting  x^  =  p,  whence  we  have  y^±a=  0. 
Solving  this  for  y  we  have  3  roots.  Call  them  a,,  Of,  a^.  Hence  to  com- 
plete the  solution  we  have  to  resolve  the  three  cubics  x^±ai=0,  x* ±a2=0, 

X^  ±  «:,  =  0. 


Exponential  Equations. 

294,  ^Exponential  Equations  are  equations  in  which  the 
unknown  quantity  or  quantities  are  involved  in  tlie  exponents. 

1 
III.    a^  +  by-  e,    a' =  d,    2'  =  43,    3*'  =  2,      y  =  256,      «*  =  100,       and 
xy  —  y  =  in  are  exponential  equations. 

2f)o»  JProb*  1, —  To  soloe  an  exponential  equation  of  the  form 

a'  —  ni. 

Solution. — Taking  the  logarithms  of  both  members  we  have  x  log  a  —  log  m 
{180 f  181)  ;  whence  x  =  ^ —  .  Therefore  finding  the  logarithms  of  m  and 
a  from  a  table  of  logarithms,  and  dividing  the  former  by  the  latter,  we  find  x. 

200,  JPvob,  2, —  To  solve  an  exponential  eqx(,ation  of  the  forin 
X*  =  m. 

Solution.— Taking  the  logarithms  of  both  members  we  have  x  log  «  =  log  w. 
Then  find  log  m  from  the  table,  and  determine  x  by  inspection  from  the  table  so 
that  X  X  log  X  shall  equal  log  m  exactly  or  approximately.* 

Examples. 
1.  Find  the  value  of  x  in  the  equation  3'  =  2546. 
80LUTI0K.    .log8  =  log2546.     ..  =  ^-^^i?  .  ?^f  =  7.138  ^ . 

2  to  6.  Solve  the  following:  (24)''=18r42;  2'=2673;  (11)''=2681; 
2^=10;     5'=:1;     (12)"  =  !. 

7.  Find  the  value  of  x  in  the  equation  r"  =  3561. 

♦  The  meth(»d  of  solving  such  equations  by  Double  Position  is  entirely  useless,  since  a  table 
of  lo;:arithniB  ia  necessary  for  that  method,  and  having  such  a  table  at  hand,  the  approximations 
can  be  made  to  any  extent  likely  to  be  desired,  more  readily  by  simple  inspccion  than  by  com- 
l)nting  the  errors  by  Double  Position.  Moreover,  the  method  here  given  affords  a:i  excellent 
exercise  in  the  use  of  the  table?. 


\ 


EXPONENTIAL  EQUATIONS.  257 

Solution.— We  have  x  log  x  =  log  3561  =  3.551572.  .  Now  looking  in  a  table 
of  logarithms,  we  soon  see  that  x  must  be  near  5,  since  5  log  5  =  5  x  .698970 
=  3.494850.  Thus  Ave  see  that  a-  >5.  Trying  5.1  we  have  5.1  log  5.1=3.608607, 
.-.  X  <  5.1.  Therefore  we  try  5.05.  5.05  log  5.05  =  3.55161955,  which  coincides  so 
nearly  with  the  required  value  of  x  log  x,  that  undoubtedly  the  lOOths  figure  is 
4.  Again,  for  a  nearer  approximation  try  5.049,  as  the  value  of  x  is  very  near 
5.05.    5.049  log  5.049  =  3.550482.     Hence  we  see  that  x  =  5.049  + . 

8  to  15.  Solve  the  following  as  above :  af  =100;  of  =  7 ;  of  =  21; 

ar^  -  402f  =  200;  3^^  +  3^  =  100;  a^  -.%  =  2b;    a'^-'  =  c;     tr^b'^' 


16  to  21.  Solve  the  following:  x^  =  f,  and  ^=y^;  x"  =  if,  and 
af^zy";  m'-' =  n,  2i\\ii  x  +  ij  =  q;  2^  3"  =:  500,  and  2x  =  dy', 
h'"-'  =  256 ;      (a'  -  2a'¥  +  h'y-'  =  {a  -  b)''  {a  +  b)-\ 


VIZ 


22.  Given  the  fundamental  formiilse  of  Geometrical  Progression, 

7       

I  =  ar''-\     and      8  =  t^:^^  ,      to      find     the     following : 

n  =  ^Qg  ^  -•  ^Q?  ^  ^  1  .  ^  ^  log  [a  -]-  (r  -  1)S]  -  log  a  ^ 

log  /•  '  log  r  ^ 

'*-log(^^-«)-log(.V-0  ^^'^""^  ^-  -]^ -^1- 

23.  Given  the  two  fundamental  formulas  of  Compound  Interest, 
viz.,  a  =  p{l  +  r )',*     and    i  =  a  —  p,    to    find    the    following : 

_   log(;j  +  t)-log/?.  _   log  ^ -log  J??  ^ 

^  -   l^fiTTf) '  ^  -     log  (1  +  r)  '        ^^^    (^  +  ^) 

^  log(;7  +  0-log;7,  ]^g    (1    ^    ^)   ^    log  ^  -  log  p  , 

?  t 

_  log  »!  -  log  {a  -  i) .  _  log  «  -  log  {a  ~  i) 

'-  log  (1  +  r)  '     log(l  +  »)-  ^ . 

Note. — Many  problems  in  Compound  Interest,  Annuities,  and  kindred  sub- 
jects are  most  expeditiously  solved  by  means  of  logarithms.  The  student  who 
has  not  a  table  of  logarithms  at  hand  may  either  omit  the  following  examples 
in  this  section,  or  content  himself  with  selecting  the  proper  formula  and  telling 
how  it  is  applied  to  the  solution  of  the  particular  example. 

24.  What  is  the  amount  of  $100  at  Hfo  annual  compound  interest 

*  This  formula  is  obtained  thus :  letting  r  represent  the  rate  for  time  1,  expressed  decimally, 
i.  e.,  if  the  rate  is  7  per  ct.,  r=.OT,  or  — -  ,  we  have  for  time  1  (as  1  year),  a=:p+pr=p{i-\-r) ; 

fortimeS,  a=;>(l  +  r)+7?r(t  +  r)=i)(l+r)z  ;  for  time  3,  a-p(l  jr)2 +pril '^r)2=:p{l-\-ryi  \  there- 
fore for  time  <,  a=p(l+r)'. 


258  ADVANCED   COUHSE   IN   ALGEBIIA. 

for  10  years?  What  if  the  interest  is  compounded  semi-annually? 
What  if  quarterly  ?  What  in  each  case  if  the  rate  is  10^  ?  If  6^  ? 
If  3^  ? 

Sug's, — We  have  a  =p{l  +  rY,  whence  log  a  =  log  p  +  t  log  (1  +  r)  =  log 
100  +  20  log  1,035,  for  interest  at  7%  compounded  semi-annually. 

25.  In  what  time  will  a  sum  of  money  double  itself  at  10^  com- 
pounded semi-annually?  At  7^  compounded  annually?  In  what 
time  triple  ?     Quadruple  ? 

SUG.    a=z2p=p{l  +ry,  whence  2  =  (1  -H r)', and  ^  =  —i^—  . 

log(l+r) 

26.  In  what  time  will  110  amount  to  $100  at  8^  compounded 
annually  ? 

'27.  What  is  the  present  worth  of  $2000  due  3  years  hence,  without 
interest,  if  money  is  worth  10^  compound  interest  ? 

SuG. — The  present  worth  is  a  sum  which,  put  at  compound  interest  at  10^, 
will  amount  to  $2000  in  3  years.  Hence  2000  =  p  (1.1)  *,  p  standing  for  present 
worth.     Whence  log  p  =  log  2000  —  3  log  (1.1). 

28.  A  soldiers  pension  of  1350  per  annum  is  5  years  in  arrears. 
Allowing  o^  compound  interest,  what  is  now  due  him  ? 

Sug's. — The  5th,  or  last  year's  unpaid  pension  has  no  interest  on  it,  as  it  is 
just  due.  The  4th,  or  next  to  the  last,  has  1  year's  interest  due,  and  hence 
amounts  to  350  (1.05) .  The  3d  year's  pension  has  2  years'  interest  due,  and  hence 
amounts  to  350  (1.05)*.  Thus  the  total  is  found  to  be  350 +350 (1.05) +  350  (1.05)- 
+  350(1.05)'  +  350(1.05)\  or  350  { 1  +  (1.05)  +  (1.05)^  +  (1.05) '  +  (1.05)*  \ 

29.  Letting  S  represent  the  amount  of  an  annuity  a,  in  arrears 
for  t  years,  compound  interest  being  allowed,   at  r^,   show  that 

r 

30.  What  is  the  present  worth  of  an  annuity  of  $200  for  7  years, 
money  being  worth  5^  compound  interest  ? 

SuG. — Evidently,  a  sum  which,  put  at  5%  compound  interest,  will  amount  to 
the  same  sum  in  7  years,  as  the  annuity  will. 

31.  Letting  P  be  the  present  worth  of  an  annuity  «,  for  time  /,  at 
r^  compound  interest,  show  that  P=  -» ^— - — --^f — .   Also,  that  if 

the  annuity  is  perpetual  (runs  forever),  P  =  -. 


EXPONENTIAL  EQUATIONS.  259 

ScG.-men  *  =  oc ,  P  =:  - .  -^.^^  =  -  .  ^j^  =  -.  a*  it  evident!,. 

should,  since  such  an  annuity  is  worth  a  present  sum  which  will  yield  aii 
annual  interest  equal  to  the  annuity. 

32.  What  is  the  present  worth  of  a  perpetual  annuity  of  $350, 
money  being  worth  Ty^^  compound  interest  ?  If  money  is  worth 
10^  compound  interest  ? 

33.  What  is  the  present  worth  of  an  annual  pension  of  $125, 
which  commences  3  years  hence  *  (first  payment  to  be  made  4  years 
hence),  and  runs  10  years,  money  being  worth  10^  compound 
interest.^ 

SuG. — Evidently,  the  difference  between  the  present  worth  of  such  a  pension 
for  13  years,  and  for  3  years. 

34.  An  annuity  a,  which  commences  T  years  hence,  and  runs  / 
years  at  r^  compound  interest,  gives 

^       a  j  (l+rr--l_(l_+_r)^l  )        «  j  .^  .    ._._  a  +  ,)-<..o  I 

AVhen    the    annuity    is    perpetual    after    the    time    T,    we    have 
P  =  ^  (1  +  r)- ''.     Student  give  proof 

35.  Two  sons  are  left,  one  with  the  immediate  possession  of  an 
estate  worth  $12000,  and  the  other  with  a  perpetual  annuity  of  1800 
in  reversion  after  7  years:  money  being  worth  5^  compound  in- 
terest, which  has  the  more  valuable  inheritance,  and  how  mucli  ? 

3G.  What  annual  payment  will  meet  principal  and  interest  of  a 
debt  of  $2000  at  8^  compound  interest  in  5  years? 

Sug's. — The  amount  of  $3000  at  8^  compound  interest  for  5  years  =  the 
amount  of  the  annuity  a  for  the  same  rate  and  time. 

37.  Show  that  if  Z)  is  a  debt  at  compound  interest  at  rfo,  h  an 
annual  payment,  and  i  the  number  of  years  required  to  liquidate 

thedebt,^^^^g^-\^fi^-^^). 
log(l  +  r) 

38.  The  debt  of  a  certain  State  is  $20,000,000,  bearing  annual 
interest  at  4^^.  A  sinking  fund  of  $2,000,000  annually  is  set  apart 
to  meet  it.  How  long  will  it  require  to  extinguish  the  debt  ?  How 
long  if  instead  of  paying  the  $2,000,000  annually  on  the  debt,  it  is 
invested  at  6^  compound  interest? 


*  An  annuity  which  commences  after  some  epecificd  time  is  said  to  be  in  reversion. 


260  ADVANCED  COURSE  IN  ALGEBRA. 

39.  A  fanner  lius  paid  !&10  per  annum  for  newspapers,  whicli  he 
considers  liave  increased  his  net  annual  income  at  least  ^.  For  10 
years  during  which  his  net  income  has  been  $500  annually,  money 
lias  been  worth  lOj^  compound  interest.  What  is  the  total  net  gain 
to  be  credited  to  his  investment  in  neAvspapers? 

40.  A  boy  commenced  smoking  when  15  years  old.  For  the  first 
5  years  he  smoked  2  5-cent  cigars  each  day.  For  tlie  next  20  years, 
3  10-cent  cigars  per  day.  Now  had  he  abstained  from  smoking  and 
invested  at  the  end  of  each  six  months  the  amount  thus  saved,  at  10^^ 
annual  compound  interest,  how  much  would  he  have  accumulated  from 
this  source  at  the  age  of  40  ? 

41.  A  man  pays  a  premium  of  45104  per  annum  on  a  life  policy  of 
*4200  for  20  years  before  his  death.  Money  being  worth  10^  com- 
})ound  interest,  does  the  insurance  company  gain  or  lose,  and  how 
much  ? 


CHAPTEK  IV. 


DISCUSSION,  OR  INTERPRETATION,  OF  EQUATIONS. 


207,  To  DisciittSf  or  Interpret ,  an  Equation  or 
an  Algebraic  Exiyression,  is  to  determine  its  significance  for 
the  various  values,  absolute  or  relative,  which  may  be  attributed  to 
the  quantities  entering  into  it,  with  special  reference  to  noting  any 
changes  of  values  which  give  changes  in  the  general  significance. 

Such  discussions  may  be  divided  into  two  classes :  1st.  The  dis- 
cussion of  equations  or  expressions  with  reference  to  their  constants ; 
and  2d.  The  discussion  of  equations  or  expressions  Avith  reference  to 
their  variables. 

The  following  principles  are  of  constant  use  in  such  discussions  :  * 

208,  JProp, — A  fraction^  when  comjyared  with  a  finite  quantity^ 
becomes  : 

*  Those  principles,  and  in  fact  most  of  this  chapter,  have  been  considered  previously,  but 
»re  collected  here  for  review  and  connected  study. 


INTERPRETATION  OF  EQUATIONS.  261 

1.  Equal  to  0,  lohen  its  numerator  is  0  and  its  denominator  finite, 
and  when  its  numerator  is  finite  and  its  denominator  oo. 

2.  Equal  to  od  ,  when  its  numerator  is  finite  and  its  denominator  0, 
and  when  its  numerator  is  go  and  its  denominator  finite. 

3.  It  assumes  an  indeterminate  form  when  numerator  and  dejunn- 
inator  are  both  0,  and  when  they  are  both  co  .* 

Dem, — These  facts  appear  when  we  consider  that  the  value  of  a  fraction  de- 
pends upon  the  relative  magnitudes  of  numerator  and  denominator. 

1.  Let  a  be  any  constant  and  x  a  variable,  then  the  fraction  -  diminishes    as 

a 

X  diminishes,  and  becomes  0  when  x  is  0.     Again,  the  fraction  -  diminishes  as 

X 
X  increases,  and  when  x  becomes  oo ,  i.  c,  greater  than  any  assignable  magni- 

a 
tude,   —   becomes  less  than  any  assignable  magnitude  or  infinitesimal,  and  is  to 

X 

be  regarded  as  0  in  comparison  with  finite  quantities.  (See  14:2  and  151^  Dem., 
and  foot-note.) 

3.  As  X  increases,  the  fraction  -  increases,  and  hence  when  x  becomes  infinite 

a 

the  value  of  the  fraction  is  infinite.     Also  as  x  diminishes  the  value  of  -   in- 

x 

creases  ;  hence  when  x  becomes  infinitely  small,  or  0,  the  value  of  the  fraction 
exceeds  any  assignable  limits,  and  is  therefore  oo . 

X 

3.  Finallv,  if  x  and  y  are  variables,  -  diminishes  as  x  diminishes,  and  increases 

y 

as  y  diminishes.    What  then  does  it  become  when  x  =  0,  and  y  =  0  ?  i.  e.,  what  is 

0  0 

the  value  of  -  ?     Simple  arithmetic  would  lead  us  to  suppose  that  -   was  abso- 
lutely indeterminate,  i.  e.,  that  it  might  have  any  value  whatever  assigned  to  it, 

0  0 

for  -  =  5,  since  0  =  5x0  =  0;  -  —7,  since  0  =  7  x  0  =  0,  etc.     But  a  closer 

0  0 

0 
inspection  will  enable  us  to  see  that  the  symbol  -  is  not  necessarily  indetermi- 
nate, or  rather  that  the  expression  which  takes  this  form  |or  particular  values  of 
its  components,  has  not  necessarily  an  indefinite  number  of  values  for  these 

X 

values  of  its  components.     Thus,  what  the  value  of  —  will  be  when  x  and  y  each 

y 

diminish    to  0  will   evidently  depend  upon  the  relative  values  of  x  and  y  at 
first,  and  which  diminishes  the  faster.     Suppose,  for  example,   that  y  —  Tix; 

X  X 

then  -  =  — .     Now,  suppose  x  to  diminish ;  the  denominator  will  diminish  5 
y        5x 


*  By  this  is  meant  that  ;:  and   —  may  have  a  variety  of  values,  not  that  they  necessarily 
0  00 


do  have. 


262 


ADVANCED  COURSE  IN  ALGEBRA. 


X       0 
or  -  =  - 

y      0 


7x      7' 


or 


times  as  fast  as  the  liuiitterAtor,  Arid  whatever  the  value  of  x,  the  value  of  the 

fraction  will  be  i.     So  if  y  =  Tar,  -  =  —  ,  which  is  \  for  any  value  of  x.    Hence 

y      7^ 

x        0         X        \ 
when  .T  =  0,  and  y  =  0,  we  have  -  =  -  =  —  =  - 

y        0        5^       5 
X       a  ... 

-  =  -  =  any  other  value  depending  upon  the  relative  values  of  x  and  y.    So, 

a;         00  iC        00  a; 

also,  if  a;  =  00 ,  and  y  =  oo ,   -  =  —  ;  but    if    y  =  Qx,  we  have  -  =  —  =  — 

y        <»  y       00        6a: 

1  a;        00         a;        1 

=  -  .    And  so  if  y  =  10^,  we  have  -  =  —  =    -— -  =  — .    Thus  we  see  that  the 
6  y        00        l(te      10 

mere  fact  that  numerator  and  denominator  become  0,  or  become  oo ,  does  not  de- 
termine the  value  of  the  fraction,  i.  e.,  gives  it  an  indeterminate  form, 

299.  A  Meal  dumber  or  Quantity  is  one  which  may  be 
conceived  as  lying  somewhere  in  the  series  of  numbers  or  quantities 
between  —  oo  and  +  oo  inclusive. 

III. — Thus, if  we  conceive  a  series  of  numbers  varying  both  ways  from  0,  i.e. 
positively  and  negatively  to  oo ,  we  have 

-4,-3,-2,-1,   0,    +1,    +2,    +3,    +  4, 


—  00    -  -   - 


+   00. 


Now  a  real  number  is  one  which  may  be  conceived  as  situated  somewhere 
within  these  limits;  it  maybe  +,  — ,  integral,  fractional,  commensurable,  or 
incommensurable.  Thus  +  15624  and  —  15624  will  evidently  be  found  in  this 
series.  +  ^i-  may  be  conceived  as  somewhere  between  +  5  and  +  6,  though  iti 
exact  locality  could  not  be  fixed  by  the  arithmetical  conception  of  discontinuous 
number.  So,  also,  —  ^3^  is  somewhere  between  —  5  and  —  6.  Again  4-  Vs  is 
somewhere  between  4-  2  and  +  3,  though,  &s  above,  we  cannot  locate  it  exactly 
by  the  arithmetical  conception. 

The  following  Geometrical  Illustration  is  more  complete  than  the  arithmetical. 
Thus  let  two  indefinite  lines,  as  CD  and  AB,  intersect  (cross)  each  other,  as  at  0, 
Now  let  parallel,  equidistant  lines  be  drawn  between  them.    Call  the  one  at  a 


"""-^  f' 

t-»-T    -•-»-•»    -Is     -1 


»--^ 


+ 1,  that  at  6  will  be  +2,  at  <;  +3,  etc.  So,  also,  the  line  at  a'  being  —1,  that  at 
b'  will  be  —2,  at  c'  —3,  etc.  Xow  conceive  one  of  these  lines  to  start  from  an 
infinite  distance  at  the  left  and  move  toward  the  right.     When  at  an  infinite 


INTERrEETATION   OF   EQUATIONS.  203 

distance  to  the  left  of  0  its  value  would  be  —  go  ,  and  in  passing  to  0  it  would 
pass  through  all  possible  negative  values.  In  passing  0  it  becomes  0  at  O, 
changes  sign  to  +  as  it  passes,  and  moving  on  to  infinity  to  the  right,  passes 
through  all  possible  positive  values.  Hence  we  see  how  all  real  values  are  em- 
braced between  —  oo  and  +  oo  inclusive  * 

300.  An  Imaginary  Wuniher  or   Quantity  is  one 

which  cannot  be  conceived  as  lying-  anywhere  between  the  limits  of 
—  GO  and  +  00 ,  as  explained  above.  The  algebraic  form  of  such  a 
quantity  is  an  expression  involving  an  even  root  of  a  negative  quan- 
tity.f     (See  Part  L,  218.) 


Examples. 

1.  What  are  the  values  of  x  and  y  in  the  expressions  x  =  — ^^^  , 

a  —  a 
aV  —  a'h      ,        ,       ,,       ,  ,     , 

y  — 3~"r-  i  when  b  =  o  and  a  and  a  are  unequal  ?     When  h^V 

and  a  —  a!'i  When  a  —  a  and  I  and  b'  are  unequal  ?  What  are  the 
8\(jns  of  X  and  ?/  when  b>h'  and  a  >  a!,  the  essential  signs  of  «,  a\ 
h,  and  b'  being  +  ?  When  h>  b'  and  a  <a'?  If  a'  and  b  are  essen- 
tially negative,  and  a  =  a',  and  b  =  b',  what  are  the  values  of  x  and 
y  ?     If  rt'  and  b'  are  each  0  ? 

f 

2.  Wliat  qeneral  relation  between  a  and  a'  renders ;  =  0  ? 

•^  1+  aa 

What  renders  it  oo  r 

SoucTiON. — To  render  z ,=  0,  we  must  have  a'  —  a  =  Q,  and  1  +  aa' 

1  +  aa 

finite  or  infinite  ;  or  else  we  must  have  1  +  a«'  =  oo ,  while  «'  —  «  is  finite  or  0 

(208).     Now  rtt'—  rt  =  0  gives  «'  =  a  ;  whence ;  =  :: —    which  is  0  for 

*  \  +  aa'      \-^a^' 

any  value  of  a  finite  or  infinite.  Hence  the  relation  a'  =  a  fulfills  the  first  re- 
quirement. Let  us  now  see  if  l-|-a«'=oo  will  also  fulfill  this  requirement.  This 
gives  aa'  =  oo ,  since  subtracting  1  from  oo   would  not  make  it  other  than  oo . 

Thus  we  have  a'  =  — .     Hence  for  all  finite  values  of  a  (including  0)  a'  is  oo , 
a 


*  For  example,  the  Ptndcnt  who  is  acquainted  with  the  elements  of  geometry  knows  how  to 
construct  a  line  which  ia  exactly  equal  to  >/5  (Geom.,  Part  I.,  110).  This  line  he  can  locate 
between  +2  and  +  3,  and  also  between  -  2  and  -  3,  since  y/b  is  both  +  and  -. 

t  Tran?cendental  functions  afford  other  forms  of  imaginary  expressions  ;  for  example, 
gin~^  2,  i«cc~'  }4,  log  (-130),  log  (-m),  etc.  But  our  limits  forbid  the  consideration  of  the  iii- 
tcrprctarioi:  of  imaginaric!*,  except  in  the  most  restricted  sense,  as  indicating  incompatibility 
with  the  arithmetical  sen^c  of  the  problem. 


264  APVANCED  COURSE  IN  ALGEBRA. 

riid .  =  — ;    =  -,  wliich  can   only  be   0   when  a  =  oo.      Therefore   the 

1  +  aa      aa  a 

particulnr  values  a  =  co  =  a  =  cc ,  render =  0  ;  but  no  genend  values 

do. 

a!  —  (i 
Again,  in   order  that j  =  cc  ,  wo  must  have   1  -+-  aa'  =  0,  and  a'  —  a 

finite  or  infinite  ;  or  else  we  must  have  a'  —  ^  =  oo ,  and  1  +  aa'  finite  or  0. 

a'+i 
T.T       .    .       ,      n     •  la—a  a'      a'^ +1      «'«  +  1 

^low  1  +  aa  =0  gives  a  = ;  ; r  = =  — = =  oo 

a      1  +  aa        ^      "'        a  —  a  0 

a' 

for  any  value  of  a!  finite  or  infinite.     Therefore  the  general  relation  a= 

a'  —  a 

between  a  and  a'  renders  ;; ,  =oo .+     Let  us  now  see  if  the  relation  a'— «  =  00 

1  +  aa  ' 

will  do  the  same.     Now  if  «'  —  a  =  00 ,  one  or  the  other  (a'  or  a)  must  be  00 . 

Let  a'=  00 .     We  then  have ,  =  — >  =  -,  which  can  only  be  00  when  a=0. 

\  -\-  a^i       aa       a 

Hence  the  particular  values  a'=  oo  and  «  =  0  render ;  =  00 ,  but  no  gen- 
eral values  meet  the  requirement  unless  a  = ;. 

3.  What  general  relation  between  a  and  a  renders  — ; =  0? 

°  a  -\-  a 

AVhat  renders  it  00  ? 


4.  In  the  expression  y  =  —  2a:  +  4  ±  ^x^  —  4.r  —  5,  how  many 
values  has  y,  in  general,  for  any  particular  value  of  x  ?  For  what 
value  or  values  of  x  has  y  but  one  value  ?  For  what  values  of  x  is  y 
real?  For  what  imaginary?  For  what  values  of  x  is  y  iH)sitive? 
For  what  negative? 

SonjTiON. — Writing  the  expression  thus,  y  —  —  (2j;  —  4)  ±  ^/x'^  —  Ax  —  5, 
we  see  that  the  value  of  y  is  made  up  of  two  parts,  viz.,  a  rational  part  — (2.t— 4), 
and  a  radical  part  \/x^  —  4x  —  5.  But  the  radical  part  may  be  taken  with 
either  the  +  or  the  —  sign.  Hence,  in  general,  for  any  particular  value  of  x 
there  are  two  values  of  y.  2d.  But  if  such  a  value  is  given  to  x  as  to  render  the 
radical  part  0,  for  this  value  of  x,  y  will  have  but  one  value,  viz.,  the  rational 
part.     But  the  condition  /y/aj'  —  4a:  —  5  =  0  gives  x  —  ^  and  —  1.      Thus  for 

*  This  redaction  i?  made  by  dropping  a  and  1,  since  the  subtraction  of  a  finite  from  an  in- 
finite, or  the  addition  of  a  finite  to  an  infinite,  does  not  change  the  character  of  the  infinite. 
Thus,  in  this  case,  to  assume  that  dropping  a  and  1  aflTected  the  relation  between  numerator  and 
denominator,  would  be  to  assign  to  a  and  1  some  values  with  respect  to  tlic  infinite  a'.  But 
this  is  contrary  to  the  definition  of  an  infinite. 

t  It  is  to  be  observed  th^t  the  relation  a  =  -  —  requires  that  a  and  a'  have  difterent  essen- 
tial signs;  while  the  relation  a'  =a  requires  that  they  have  the  same  essential  signs. 


INTERPRETATION  OF  EQUATIONS.  265 

i*  =  5,  y  =  —  C,  but  one  value  ;  and  for  a;  =  —  1,  y  =  +  C,  also  but  one  value, 
od.  To  ascertain  for  what  values  of  x,  y  is  real,  we  observe  that  y  is  real  v/hen 
x^  —  4»  —  5  is  positive,  and  imaginary  when  x^  —  4.t;  —  5  is  negative.  Now 
for  X  positive  x^  —  (4c  +  5)  is  -f-  when  x^  >  4j  +  5  ;  and  for  x  negative,  we 
have  05*  -H  4e  —  5,  which  is  positive  when  x^  4-  4.c  >  5.  The  former  inequality 
gives  a?'  —  4«  -F  4  >  9,  or  x  >  ^  \  and  the  latter  gives  a;^  +  4i»  +  4  >  9,  or  .^  >  1. 
Hence  for  positive  values  of  x  greater  than  5,  y  is  real,  and  for  negative  values 
of  X  numerically  greater  than  1,  y  is  real.  The  4tli  inquiry  is  answered  by  this: 
y  is  imaginary  for  all  values  of  x  between  —1  and  +5.  5th.  To  ascertain  what 
4-  values  of  x  render  y  +,  and  what  — ,  we  observe  that  —  (3a;— 4)±  y^a?^^^^^ 
can  only  be  4-  when  the  +  sign  of  the  radical  part  is  taken  and  when 
^x^  —  4aj  —  5  >  3a;  —  4.  This  gives  a;  <  3  ±  ^—  3,  t.  e.,  an  imaginary 
quantity.  Hence  y  is  never  +  for  a;+.  Taking  the  negative  sign  of  tlit; 
radical  we  see  that  both  parts  of  the  value  of  y  are  — ,  and  consequently  y  is  ^ 
real  and  negative  for  all  +  values  of  x  which  render  y  real,  i.  e.,  for  values 
greater  than  5.  Finally,  for  x  —  we  have  y  =  3a;  +  4  ±  y^a;^  -f-  4a;  —  5.  Now 
when  we  take  the  +  sign  of  the  radical  both  parts  are  + ;  hence  this  value  of 
y  is  always  +.  When  we  take  the  —  sign  of  the  radical  y  is  negative  if 
2a;  +  4  <  \/x^  +  4a;  —  5.  But  this  gives  a;  <  —  3  ±  '\/—  3.  Hence  y  is  never 
negative  for  any  negative  value  of  x.  Therefore  both  values  of  y  are  positive 
and  real  for  all  negative  values  of  x  numerically  greater  than  1. 

5  to  22.  Discuss  as  above  the  values  of  ?/  iu  the  following ;  i.  e., 
Ist  Show  how  many  values  y  has  i)i  geiieral,  and  whether  they  are 
equal  or  unequal ;  2(1.  For  what  particular  value  or  values  of  x,  y 
has  but  one  value  ;  3d.  For  what  values  of  x,  y  is  real,  and  for  what 
imaginary  ;  4th.  For  what  values  oi  x^y  \B  +,  and  for  what  —  ;  5th, 
Also  determine  what  values  of  x  render  y  infinite ; 


(5.)  y'  +  2xy  -  2:r'-  -  4^  -  a;  +  10  =  0;  * 

(6.)  y  —  'Zxy  +  2:c'  —  2?/  4-  2.r  =  0 ; 

(7.)  y'  +  2xy  +  x'  -  6?^  +  9  =  0 ; 

(8.)  y""  +  2xy  +  3./;*  -  Ax  ^  0  ; 

(9.)  y'  -  2xy  +  3.6-^  +  2y  -^  Ax  --3  =  0; 
(10.)  y'  +  2xy  -^  3^-^  -  4.c  -  0 ; 
(U.)  y'  —  2xy  +  x^  -^  xz:zO; 
(12.)  y'  -  2xy  +  x'  -  4y  ^  x  -^  A  -  0-, 
(13.)  /  -^  2xy  +  ar*  -f  2.y  4-  1  =  0 ; 
(14.)  f  -  2x'  -  2y  +  6a:  -  3  =  0 ; 
(15.)  f  -  2xy  -  ix'  -  2?/  +  7a;  --  1  =  0; 
(IG.)  y'  ^2xy-2^0; 
(17.)  y'  -  2xy  +  2?/  4-  4.?;  -8  =  0; 


♦  In  all  cases  solve  the  eqnation  for  y  in  the  first  place.    In  this  exaropl© 


SI66  ADVANCED   COURSE  IN  ALGEBllA. 

(18.)  Af  +  4:x'  -h  2y  -  3a;  +  12  =  0; 

(19.)  Sy'-S3^=12; 

(20.)  12/  + 4a:*  =  20; 

(21.)  x'-^/=16; 

(22.)  x'  -y'  =  20. 

23.  Discuss  the  equation  ay^  —  x?  +  {h  —  c)  a:'  +  hex  =  0,  as  above, 
■when  ^  >  c ;  also  when  a  >  b. 


SUG 


'8.    y  =  ±  -7  \^x'^  —  {b  —  c)x*—  hex.  Whence  we  see  that  y  has  two  values 
a* 

for  every  value  of  x,  numerically  equal,  but  with  opposite  signs,  y  is  0,  when 
ar'  —  (&  —  e)x*  —  bcx  =  0 ;  t.  e.,  when  a;  =  0,  ar  =  6,  and  —  c.  Again  1/  is  real  for 
z  +,  when  a;-*  >  (6  —  c)a;*  +  ftca;,  or  a:*  >  (b  —c)x  +bc;  which  gives  x  >  b.    For 


a?—,  we  have  y  =  ± —r^^— x^  —  {b  — c)x* -\-bcx,   which   gives  y   real   when 

a?'  +  (6  —  c)  a;-  <  bcx,  which  gives  x  numerically  leas  than  c,  i.  c,  greater  than 

—  c.  Hence  y  is  imaginary  for  all  values  of  x  between  0  and  +  b,  and  real  for 
all  values  of  x  from  +  6  to  +qo  .     So  also  y  is  real  for  all  values  of  x  from  0  to 

—  c,  and  imaginary  for  all  values  of  x  from  —  c  to  —  oo . 

X  —  b 

24.  Discuss  us  above  y^  =  {x  —  a)* ,  showing  that  in  general 

X 

y  has  two  values  numerically  equal  but  with  opposite  signs  ;  that  it 
is  0  for  X  =  «,  and  x=  b\  is  imaginary  from  x  =  Q  io  x=.h  (except 
when  x=z  a,  b  being  greater  than  a) ;  real  from  x  =  b  to  a:=  +  <», 
aud  rcal  for  all  negative  values  of  x,  i.  e.,  from  a:  =  Otoa;=  —  00; 
and  that  for  a:  =  0,  y  =  ±  go  ,  and  for  x=.  -^cOf  y  =  zh  00  ;  also  for 
x=  —  00,  y  =  -±1  cc. 

25.  Show  from  the  equation  y  +  x^y  =  x,  that  y  =  0  when  a;  =  0, 
+  00,  and  —00  ;  also  that  y  has  but  one  value  for  any  particular 
value  of  a:;  that  it  is  -f  when  x  is  +,  and  —  when  a:  is  —  ;  and  that 
y  increases  numerically  as  x  pjisses  from  0  to  +l,and  from  0  to  —1, 
but  that  it  diminishes  numerically  as  x  passes  from  +  1  to  -f  00 , 
and  filso  from  —  1  to  —  oo . 

26.  Discuss  y^x  =  4a'  (2a  —  x)  with  reference  to  y  as  a  function 
of  X,  as  above. 

27.  Show  that  in  the  equation  y^  —  3axy  +  x^  =  0,  y  has  tliree 
real  values  between  the  limits  x  =  0,  and  x  =  a\/^,  and  only  one 
real  value  between  the  limits  x  =  a\/  4:  and  a;  =  +  qo  ,  and  also  be- 
tween the  limits  a:  =  0  and  a;  =  —  00 . 

Sue. — This  is  done  by  means  of  Cardan's  formula.  (See  280, ) 


INTERPllETATION   OF    EQUATIONS.  267 

301,  Arithmetical  Interpretation's  of  Negative  axj) 
Imaginary  Solutions. 

1.  A  is  20  years  old,  and  B  16.  When  will  A  bo  twice  as  old 
asB? 

SuG's.— We  have  20  +  ic  =  2  (16  +  «) ;  whence  x  —  —  12.  The  arithmetical 
interpretation  of  this  result  is  that  A  will  never  be  twice  as  old  as  B,but  that  he 
teas  twice  as  old  12  years  ago,  i.  e.,  when  he  was  8  and  B  4. 

2.  A  is  rt  years  old,  and  B,J.  When  will  A  be  n  times  as  old  as 
B  ?  For  71  >  1  what  are  the  possible  relative  values  of  a  and  h  con- 
sistently with  the  arithmetical  sense  of  the  problem?  Interpret  for 
a  >  nh,  a  =  nh,  a  <  nh  when  n  >  1.  Also  for  ?i  =  1,  «  >  nb,  a  <  nh, 
and  a  =  nh. 

3.  Two  couriers,  A  and  B,  are  traveling  the  same  road  in  the 
same  direction,  the  former  at  rate  «,  the  latter  at  rate  K  They  are 
at  two  places  c  miles  apart  at  the  same  time.  Where  and  when  are 
they  together  ? 

Solution  and  Discussion. — Let  XY  represent  the  road  which  the  couriers 
are  traveling  in  the  direction  from  X  to  Y,  and  A  and  B  the  stations  which  they 


t 


pass  at  the  same  time,  A  being  at  A  when  B  is  at  B,  and  D  or  D'  the  place  at 
which  they  are  together.  Call  the  distance  from  B  to  the  place  at  which  they 
are  together  ±x,  +  x  when  D  is  beyond  B,  and  —x  when  it  is  on  the  hither 
side  of  A  and  B,  as  at  D'.  Then  the  distance  from  A  to  the  point  at  which  they 
are  together  is  c  4-  (±  a;).  Now  disregarding  the  essential  sign  of  x,  and  leaving 
it  to  be  determined  in  the  sequel,  we  have 

Distance  A  travels  from  kr=c  -\-  x, 

Distance  B  travels  from  B  =        x; 

Time  from   passing  A  and  B  to  the  time   they  are  together  —^  and  - . 

But  these  are  equal.     Hence  we  are  to  discuss  the  equation 

C  A-  X       X                      ^c            ,       .               oc 
^  ^     =  _  ,  or  a;  = r  ,  and  c-\-x=     


a         b'  a-b  a  -b 

The  points  to  be  noticed  in  the  discussion  are,  (1)  when  a>b,  (2)  when  a  <h, 
(8)  when  a  =  b,c  being  greater  than  0  in  each  case  but  not  co .  Also  the  like 
cases  wlien  c  =  0. 

W?ien  c  >  0  but  riot  oo . 

We  have,  for  a  >  b,x  positive,  which  shows  that  the  point  at  which  they  ar# 


i:68  ADVANCED  COtRSE  IN  ALGEBRA. 

together  is  at  the  right  of  B,  i.  e.,  in  the  direction  which  they  are  travelmg. 
The  time,  r  ( or  — ^ )>  ^^  positive,  which  shows  that  they  are  together  qfter 
passing  A  and  B. 

For  a  <b,  x  is  negative,  and  c  -h  x,  which  equals  j ,  is  also  negatire. 

This  shows  that  they  were  together  at  a  point  at  the  left  of  A,  that  is,  hefore 
they  reached  the  stations  A  and  B.     With  this  the  expressions  for  the  time  also 

agree.     Thus  r  becomes  —  j- ,  and is  also  negative,  since  in  this  case  x>c. 

0  0  d 

MTi  ».  he        he  -,  ae        ac  ,.,.,. 

When  a  =  0,x— r  =  —  =oo ,  and  c  +  a?  = j-  =  -77-  =ao  ;   which  mdi. 

a—b       0  -     a—ft       0 

cates  that  they  are  never  together. 

When  c  =  0. 

In  this  case  x  = r  =  0,  and  c  +  a?  = r  =  0,  for  a  and  6  unequal,  indi- 

a  —  b  a  —  b        *  ^ 

eating  that  they  are  together  when  they  are  at  A  and  B.     This  is  evidently  cor- 

bc         0 

rect,  since  A  and  B  coincide   in   this   case.     When  a  =zb,  x  — =  -  ,  and 

a  —  b      0 

c  -f  a;  =  — ,  which  shows  that  they  are  always  together,  -  being  a  symbol  of  iu- 

determination  which  in  this  instance  may  have  any  value  whatever,  as  we  see 
from  the  nature  of  the  problem. 

302,  ScH. — The   student   should    not  understand  that  the  symbol  - 

dlway»  indicates  that  the  quantity  which  takes  this  form  has  an  indefinite 
number  of  values.  It  is  frequently  so,  but  not  necessarily.  The  indeter- 
mination  may  be  only  apparent^  and  what  the  value  of  the  expression  is 
must  be  determined  from  other  considerations.  The  Calculus  affords  the 
most   elegant   general  methods  of  evaluating  such  expressions.     But  the 

simple  processes  of  Algebra  will  often  suffice.     Thus  for  a;  =  1,  -r— - —  =  -. 

1  —  a;^  \  —  x^ 

But  -zr =  1+  a;  +  a;*,  which,  for  a;  =  1,  is  3.     Hence  -z =  3,  for  a;=l. 

1  —X  '  '  '  1  — a;        ' 

Here  the  apparent  indetermination  arises  from  the  fact  that  the  particular 

assumption  (that  a;  =  1)  causes  the  two  quantities  between  which  we  wish 

the  ratio,  viz.,  the   numerator   and  denominator,  to  disappear.      Let  the 

1  _  3.5 

student  find  that  z ^ j  =  2^  for  x  =  \,     (See  also  298,  3d  part  of 

A — aj-j-a?  "~"a5 

demonstration.) 

4.  Two  couriers  starting  at  the  same  time  from  the  two  points 
A  au(l  B,  c  miles  apart,  travel  toward  each   other  at   the   rates  a 


INTERPRETATION   OF   EQUATIONS.  269 

and  h  respectively.    Discuss  the  problem  with  reference  to  the  place 
and  time  of  meeting.     (Consider  when  a>  h,a<h,  and  a  =  J.) 

5.  Two  couriers,  A  and  B,  are  traveling  the  same  road  in  the 
same  direction,  the  former  at  rate  «,  and  the  latter  n  times  as  fast. 
They  are  at  two  places  c  miles  apart  at  the  same  time.  Discuss  the 
problem  with  reference  to  place  and  time  of  meeting  as  in  Ex.  3, 
adding  the  considerations,  w  >  1,  n  <1,  n  =1,  n  —  0. 

6.  Divide  10  into  two  parts  whose  product  shall  be  40. 

Solution  and  Discussion. — Let  x  and  y  be  the  parts,  then  a;  +  y  =  10, 
xy  =  40,  and  x  =  5  ±  -y/—  15,  y  =  5  T  \/—  15.  These  results  we  find  to  be 
imaginary.  This  signifies  that  the  problem  in  its  arithmetical  signification  is 
impossible  :  this  indeed  is  evident  on  the  face  of  it.  But,  although  impossible 
in  the  arithmetical  sense,  the  values  thus  found  do  satisfy  the  formal,  or  alge- 
braic sense.  Thus  the  sum  of  5  +  /y/—  15  and  5  —  ^y/—  15  is  10,  and  the 
product  40. 

7.  Tlie  sum  of  two  numbers  is  required  to  be  a,  and  the  product 
b:  what  is  the  maximum  value  of  b  which  will  render  the  problem 
possible  in  the  arithmetical  sense?  What  are  the  parts  for  tliis 
value  of  Z*  ? 

8.  Divide  a  into  two  parts,  such  that  the  sum  of  their  squares 
shall  be  a  minimum. 

Suq's. — Let  X  and  a—x  be  the  parts,  and  m  the  minimum  sum.    Then 
x*  +  (a  —  xY  =  2x^  —  2ax  +  a^  =m; 

whence    x  =  ^a  ±  ^  '\/  2m  —  a^.      From   this   we  see  that  if  2m  <  a^,  x  is 
imaginary.     Hence  the  least  value  which  we  can  have  is  2m  =  a^^  or  m  =  \a^. 

9.  Divide  a  into  two  parts,  such  that  the  sum  of  the  square  roots 
shall  be  a  maximum. 

10.  Let  d  be  the  difference  between  two  numbers :  required  that 
the  square  of  the  greater,  divided  by  the  less,  shall  be  a  minimum. 

11.  Let  a  and  h  be  two  numbers  of  which  a  is  the  greater,  to  find 
a  number  such  that  if  a  be  added  to  this  number,  and  h  be  sub- 
tracted from  it,  the  product  of  this  sum  and  this  difference,  divided 
by  the  square  of  the  number,  shall  be  a  maximum. 

Sug's.— Let  n  be  the  number,  and  m  the  required  maximum  quotient.     Then 

n^  -\-(a  —  b)n  —  ah  .  ^    , 

by  the  conditions ^ ^^^ -  m,  whence  we  fand 


n' 


a-b         \/a^  -h  2«*  +  6*  —  ^<tbm 


2(1  -  w)  2(1  -  m) 


2T0  ADYANCED  COURSE  IN  ALGEBRA. 

From  this  we  see  tliat  tLe  greatest  value  which  m  can  have  and  render  n  real 

iam  =  — J—-  .    This  gives  n  =  —  — ; r  = r  . 

4ab  ^  2(1— m)       a  —  b 

12.  To  find  the  point  on  a  line  passing  through  two  lights  at 
which  the  illumination  will  be  the  same  from  each  light. 

Solution. — Let  A  and  B  be  the  two  lights,  and  XY  the  line  passing  through 

X if 6 ^1 f -V 

them.  Let  a  be  the  intensity  of  the  light  A  at  a  unit's  distance  from  it,  6  the 
intensity  of  B  at  a  unit's  distance  from  it,  c  the  distance  l)etween  the  two  lights, 
as  AB,  and  x  the  distance  of  the  point  of  equal  illumination  from  the  light  A,  as 
AD  (or  AD).  Then,  as  we  learn  from  Physics  that  the  illuminating  effect  of  a  light 
varies  inversely  as  the  square  of  the  distance  from  it,  we  have  for  the  illumina- 
tion of  the  x)oint  D  by  light  A  -^  ,  and  for  the  illumination  of  the  same  point 

by  light  B, r,  .     But  by  the  conditions  of  the  problem  these  effects  are 

(c  —  X) 

equal ;  hence  we  have  the  equation  to  be  discussed ;  viz.. 


{c  -  X)* 
This  gives  ^ j —  =  -;  or 


(c  —  x)*       b  e  —  x  .fb         ±  a/& 

c      ^       ±  \fb          c        \/  a  ±  \/~b 
or  1  = ^  ;  or  -  =  ~ ^ —  ; 

or,  finally,  x=.c — z=^ r,  and  a?  =6 — z^ -, 

wliich  are  the  values  of  ar  to  be  discussed. 
Discussion. — I.  Let  c  be  finite  and  >  0. 

1.  When  a>b,  x=e       J —  >ic,  since  "^  ^        >  ^  for  ^  >  &.     Tliis 

is  as  it  should  be,  since  for a>  b  the  point  of  equal  illumination  will  evidently 
be  nearer  to  B  than  to  A.  Again,  the  other  value  of  x  gives  x  =  c  — ^ r  >  c, 

-' — is  +  and  >  1,  when  a>  b.     Hence  we  learn  that  there  is  a 


point  beyond  B,  as  at  D',  where  the  illumination  is  the  same  from  each  light. 

If  we  assume  y^=2>^/T,  AD  =  ^  c,  and  AD'  =  2c. 

2.  It  is  evidently  unnecessary  to  consider  the  case  when  rt  <  ft,  since  this  Avould 
only  situate  the  points  of  equal  illumination  with  reference  to  A  as  the  j)reced- 
ing  discussion  does  with  reference  to  B. 


INTERPRETATION    OF   EQUATIONS.  271 

3.  When  a  =  b,x  =  c-^^^^^^  =  j^c,  since 'S^-^=  ^  =l  This 

is  as  it  should  be,  since  it  is  evident  that  in  this  case  the  point  of  equal  illumina- 
tion is  midway  between  the  lights.     Again,  for  tlie  second  value  of  w,  we  have 

A/a 

x  =  c  — — —  =  GO ,     This  is  also  evidently  correct ;  for  when  the  lights  ar« 

^J a—  ^  b 
of  equal  intensity  there  can  i>e  no  point  beyond  B,  for  example,  at  which  the  illu- 
mination from  A  will  be  equal  to  that  from  B,  except  wlien  a;  =  co ,  for  which 
the  illumination  is  0  for  each  light,     [Let  the  student  give  the  reason,] 

II.   When  c  =  0     In  this  case  the   original  equation  —x  =  ;; ^-r  becomes 

-J  =  -^  ,  whence  a—b.    We  then  have  x=c  — ^ —  =  0  ;  and  x  =c  — ~— — :::. 

^        ^  ^a+^b  \^a—^/b 

c-v/  a  0 

=  — ^rr^ —  =  -.     The  former  shows  that  there  is  a  point  of  equal  illuniina- 

^  a-^yb      0 
tion  where  the  lights  are  (when  c  =  0  they  are  together),  and  the  latter  shows 
that  any  point  in  the  line  is  equally  illuminated  by  each  light.     Both  these  con- 
clusions are  evidently  correct  * 


•  In  discussing  this  problem,  some  have  committed  the  error  of  considering  Hint,  8ince  fo 

c  =  0  and  a  and  b  unequal,  x  =  c  — — r  =  0,  therefore  there  i?  a  point  of  equal  illumi  nation 

\^ a±  \^b 
at  the  i)oint  where  the  lights  are  situated !    This  is  evidently  absurd,  since  the  hypothesis  is 
that  the  lights  are  of  uneqiiul  intensitj'.      The  error  consists  in  not    perceiving  that   the 

hypothesis,  c  -0,  excludes  the  hypothesis,  a  and  h  unequal.  That  the  hypotheses  a  ^  b  are 
excluded  by  the  hypotheses  <^=  0  and  that  there  ia  a  point  of  equal  illumination,  is  self-evident. 
Perhaps  the  student  may  think  that  these  conditions  are  no  more  inconsistent  than  those  in  I.  %, 
above,  viz,,  c  finite,  a=ft,  and  a  point  of  equal  illumination ;  and  that,  if  in  the  former  case  we  in- 
terpret a  =  c '^--  =  00  as  indicating  a  point  of  equal  illumination  at  a^  =  oo,  we  should  in 

^ a-  ^  b 

this  interpret  x  = M— -r  =  0  as  indicating  a  point  of  equal  illumination  at  the  place 

where  the  light?  are  situated.    But  the  closing  remark  in  I,  3  will  clear  up  this  difficulty. 


ilr, ^ 

PPJ^IVDIX 


SECTION  L 

SERIES. 

303 •  A  Series  is  a  succession  of  related  quantities  each  of 
which,  except  the  first  or  a  certain  number  of  the  first,  depends  upon 
the  next  preceding,  or  a  certain  number  of  the  next  preceding, 
according  to  a  common  law.  Each  of  the  quantities  is  called  a 
Term  of  the  Series. 

III. — A  Progression,  as  1,  3,  5,  7,  etc.,  or  3,  6,  12,  24,  etc.,  is  a  series  in  which 
each  tenn  after  the  first  depends  upon  the  next  preceding  according  to  a  common 
law.  The  numbers  1,  3,  7,  11,  21,  39,  71,  131,  etc.,  constitute  a  series  in  which 
oach  term  after  the  third  is  the  »um  of  the  three  next  preceding.  The  numlK;rs 
2,  3,  5, 17,  88,  1513,  etc.,  constitute  a  series  in  which  each  term  after  the  first 
three  is  the  product  of  the  two  next  preceding  +  the  third  preceding. 

304.  A  Mecurring  Series  is  a  series  in  which  each  term 
after  the  first  n  is  equal  to  the  sum  of  the  products  of  each  of  the  n 
preceding  terms  multiplied  resiTCctively  by  certain  quantities  which 
remain  the  same  throughout  the  series.  These  multipliers  with 
their  respective  signs  constitute  the  Scale  of  Relation. 

III.  1,  4c,  9j;*,  16j;',  etc.,  is  a  recurring  series  whose  scale  of  relation  is 
x'\  -3jj*,  3.r,  since  {I  x  x*)  +  (4e  x  [  -  3:r*])  +  (9x*  x  3.r)  =  lCx^  The  next 
term  after  16.i; '  would  be  (4c  x  z*)  +  (9x*  x  [  -  3a;'])  +  (16a;'  x  ar)  =  25a;\ 
The  next  would  be  36a;' . 

305.  An  Infinite  Series  is  one  which  has  an  infinite  number 
of  terms.  Such  a  series  is  said  to  be  Convergent  when  the  successive 
terms  decrease  according  to  such  a  law  as  to  make  the  sum  finite  ; 
otherwise  it  is  called  Divergent. 

III.  -^,  T*^,  j-^a,  jji^-fra,  etc.,  to  infinity,  is  an  infinite,  converging  series 
whose  sura  is  ^,  That  -?,-,  +  joff  +  to^itt  +  ToJinr  +  etc.,  to  infinity  *=  i  is  evi- 
dent, since  by  division  we  liave  ^  =.3333  +  =  -i%  +  too  +  ro^oo  +  etc. 


*  The  expression  "  to  infinity  "  is  usually  omitted,  as  being  sufficiently  indicated  by  "  etc.;  ** 
and.  In  fact,  either  the  +  sign  at  the  end  or  the  "  etc."  may  be  omitted. 


SERIES.  273 

306.  To  Hevevt  a  Series  involving  an  unknown  quaufcity 
is  to  express  tlie  value  of  that  unknown  quantity  in  terms  of 
another  quantity  which  is  assumed  as  the  sum  of  the  first  series,  or 
as  involved  in  that  sum.  Thus  the  general  problem  is,  having 
given /(y)  =  ax-\-  hx"  +  ex"  +  etc.,  to  express  x  in  terms  of  y,  i,  e., 
to  find  ic  =/(?/). 

III.— Tims  to  revert  the  series  x  4-  Saj^  +  ^x^  +  7a;*  +  Oa;'  +  etc.,  is  to  express 
the  value  of  x  in  another  series  involving  y  when  g^  =  x  +  3aj*  +  5a;^  +  7a;' 
+  9a;«  +  etc.,  or  when  1  -  2y  +  5y-'  =  a;  +  3a;=*  +  5.c»  +  7a;*  +  9a;'*  +  etc.,  etc. 

307.  TJie  First  Order  of  Differences  of  a  series  is  the 
series  of  terms  obtained  by  subtracting  the  1st  term  of  the  given 
scries  from  the  2d,  the  2d  from  the  3d,  the  3d  from  the  4tli,  etc. 
The  Second  Order  is  obtained  from  the  first  as  tlie  first  is  from  the 
primitive  series.  The  Third  Order  is  obtained  in  like  manner  from 
the  second ;  etc. 

These  several  series  are  called  the  Successive  Orders  of  Differences. 

III. — Having  the  series 

1,        8,        27,        64,        125,    etc.,  we  obtain 

1st  order  of  diff's,  7,        19,        37,        61,    etc., 

2d      "       "      "  12,       18,        24,    etc., 

Cd     "       "      *'  6,  6,    etc., 

4th    "      "      "  0,    etc. 

308.  Interpolation  is  the  process  of  finding  intermediate 
functions  between  given  non-consecutive  functions  of  a  series, 
Avithout  the  labor  of  computing  them  from  the  fundamental  formula 
of  the  series. 

III. — The  logarithms  of  the  natural  numbers  1,  2,  3,  4,  5,  0,  7,  8,  etc.,  con- 
stitute a  series  of  functions.  Now  knowing  these,  interpolation  teaches  how  to 
find  intermediate  logarithms,  as  log  4.3,  4.5,  4.6,  etc.,  or  2.7,  2.72,  3.102,  7.025, 
etc.,  without  the  labor  of  computing  them  from  the  fundamental  formula  of 
the  series  {192). 

[Note. — The  student  must  guard  against  the  notion  that  every  series  is  a 
recurring  series.  Any  snccesnon  of  numbers  related  to  each  other  by  a  common 
law,  as,  for  example,  the  logarithms  of  the  natural  numbers,  is  a  series,  as  well 
as  the  more  simple  arithmetical,  geometrical,  and  other  recurring  successions.] 

300.  Some  of  the  more  important  problems  concerning  infinite 
series  are :  To  find  the  scale  of  relation  of  a  series;  To  find  the  nth 
(any)  term  of  a  series ;  To  determine  whether  a  scries  is  convergent 
or  divergent ;  To  find  the  sum  of  a  convergent  series,  or  of  n  terms 
of  any  series ;  To  revert  a  series ;  and,  To  interpolate  terms  between 


274  APPENDIX. 

given  terms.  To  these  problems  we  shall  give  attention  after 
having  demonstrated  the  following  lemm*,  which  is  of  use  in  the 
solution  of  several  of  them. 


310,  Lemma, —  The  first  term  of  the  nth  order  of  differences 

,      n(n-l)        n(n-l)(n-2)  ^       ,         ,  . 

tsa  —  nb  -\ — ^—- -c ^ -.^^^ ^  d  +  etc,  ir/ten  n  is  even,  and 

n(n  — 1)       n(n  — l)(n  — 2),       ,         ,  .      ,, 

— a  -f  nb ^c-l — ^^ ,^ -a— etc., when  n  u  odd;  a, 

b,  c,  d,  etc.,  being  successive  terms  of  the  series. 

Dem. — Letting  a,  b,  c,  d,  e,  f,  etc.,  be  the  series,  we  have 

1st  Order  of  diflf's,  h  —  a,  c  —  h,  d  —  c,  e  —  d,  f—  e,  etc., 

2d      "       "     "        c-2b  +  a,  d— 2c  ■hh,e  —  M->rC,f—2e  +  d,  etc., 

3d      "       "     "        d  — 3c  +  36  — a,  e  —  3(i  +  3c  — 6,/— 3e  +  3rf— c,  etc., 

4th     "       "     "        g  _4^  +  6c  —  46  +  a,/— 4c  +  6rf  — 4c  +  &,  etc., 

5th    "       "     "       /-  5c  +  \M  -  10c  +  56  -  a,  etc. 

Now  by  inspection  we  observe  that,  numerically,  the  coefficients  in  these 
terms  follow  the  law  of  the  coefficients  in  the  development  of  a  binomial.  Thus 
the  coefficients  in  any  term  of  the  2d  order  of  differences,  as  in  c~2b  +  a,  are 
the  same  as  in  the  square  of  a  binomial ;  those  in  any  term  of  the  3d  order,  as 
in  d—3c  -f  36— a,  are  the  same  as  in  the  cube  of  a  binomial,  etc.  Hence,  revers- 
ing the  order  of  the  simple  terms  in  the  first  terms  of  the  successive  orders,  and 
representing  the  first  term  of  the  first  order  by  i), ,  the  first  term  of  the  2d  order 
by  Di,  the  1st  term  of  the  3d  order  by  Dj,  etc.,  we  have,  for  the  even  orders, 

Bi  =a  —  2b  +  c, 

D4  =  a  —  4b  +  Qc  —  4d  +  e. 

Hence,  by  induction,  we  have,  for  the  1st  term  of  the  nth  order,  when  n  is  even, 

^  ,       n(n  —  1)         n{n  —  l){n  —  2)  ^ 

Bn  =a-nb  +  -^ — '-c p^ '  d  +  etc. 

Again,  for  the  odd  orders,  we  have 

Bx  =  —  a  +  b, 

i>3  =z  -  a  4-  36  -  3c  h  d, 

B^  =  -  a  +  hb  -  10c  +  lOrf  -  5c  +  /. 

Hence,  by  induction,  when  n  is  odd,  the  first  term  of  the  7ah  order  is 

-r.  .       n(n-\)         7i(n  —  1)  {n  -  2)  ^ 

Bn=  —  a  +  nb —- — -  c  +  -^ 4 d  -  etc.* 


*  The  author  does  not  deem  it  expedient  to  take  the  time  and  ej)ace  to  demonHtrate  more 
rigorously  thii?  law  ;  nor  does  he  fully  sympathize  with  the  idea  thai  induction  is  in  no  case  n 
eatiefactory  mathematical  argument. 


SERIES.  275 

311,  Cov, — It  will  be  observed  that  in  order  to  find  the  \st  tenn 
of  the  first  order  of  differences,  we  must  have  2  terms  of  the  series 
given  ;  to  find  the  \st  term  of  the  2d  order,  3  terms  ;  to  find  the  \st 
term  of  the  3c?  order,  4  terms  ;  and,  in  general,  to  find  the  \st  term 
of  the  nth  order  we  must  know  n  +  1  terms  of  the  series. 

Examples. 

1.  Find  the  1st  term  of  the  3d  order  of  differences  in  the  series 
7,  12,  21,  36,  62,  etc.     Also  the  1st  term  of  the  4th  order. 

Sug's. — For  the  3d  order  we  have 

i)3  =  -  a  +  3&  -  3c  +  <?  =  -  7  +  3   12  -  3  •  31  +  30  =  2. 
For  the  4th  order, 

D4  =  a  -  46  +  6c  -  4rZ  +  g  =.  7  -  4  •  12  +  6  21  -  4  •  36  +  62  =  3. 

2  to  6.  Find  the  first  terms  of  the  orders  of  differences  specified  in 
tlie  following : 

(2.)  2d,  3d,  and  4th,  in  1,  8,  27,  64,  125,  etc. 

(3.)  3d,  and  5th,  in  1,  3,  3'^  3^  3^  3',  etc. 

(4.)  5th,  in  1,  ^,  i,  J,  ^\,  ^\,  etc. 

(5.)  5tli,  in  1,  6,  21,  56,  126,  252,  etc. 

(6.)  6th,  in  3,  6,  11,  17,  24,  36,  50,  etc. 


312,  Proh,  1, — To  fiiid  the  Scale  of  Relation  in  a  recurring 
infinite  series  when  a  sufficient  number  of  terms  is  given. 

Solution. — 1st.    When  each  term  after  tJie  first  depends  on  the  next  preceding. 

— Let  m  represent  the  scale  of  relation.    Then  h  —  ma  (,304).    Whence  7n  =  -. 

a 

2d.  When,  each  term  after  the  first  two  depends  on  the  two  terms  next  preceding 
it. — Letting  m,  n  be  the  scale  of  relation,  we  have  c=ma  +  nh,  and  d=7nb  +  nc 

{304).     Whence  m  = z^,  and  n  = r-^. 

ac  —  b^  ac  —  h^ 

3d.  When  each  term  after  the  first  tJiree  depends  on  the  three  terms  next  preced- 
ing it. — Letting  m,  n,  r  represent  the  scale,  we  liave  d  =  ma  +  nh  +  re,  e—mb 
+  nc  +  rd,  and/=  mc  -\-  nd  -^^  re.  From  these  three  equations  the  values  of 
m,  n,  and  r  can  be  found. 

4th.  We  can  evidently  proceed  in  a  similar  manner  when  the  dependence  h 
upon  any  number  of  preceding  terms. 

313,  Sen. — In  applying  this  method,  if  wc  assume  tliat  tlie  dcpendeneo 
is  upon  more  terms  than  it  really  is,  one  or  more  of  the  terms  of  the  scale 
will  reduce  to  0.     If  we  assume  the  dependence  to  be  on  too  few  terms,  the 


2fl5  APPENDiy. 

error  will  appear  in  attempting  to  apply  the  scale  when  found.  If  we 
attempt  to  apply  the  method  to  a  series  which  is  not  recurring,  the  error 
will  appear  in  the  form  which  the  scale  assumes,  or  when  we  attempt  to 
apply  it. 

When  the  dependence  is  upon  two  terms,  any  two  equations  of  the  scries 
c  =  vin  +  Tib,  d  =  mh  +  Jic,  e  =  mc  -h  nd,  f—  md  4-  nc,  etc.,  will  give  the  mme, 
cnlues  for  m  and  u.  So  also  if  the  dependence  is  upon  three  terms,  any  three 
equations  of  the  series  d  =  ma  +  nb  +  rc^  e  =  mh  +  nc  +  rd,  f=  mc  +  nd  +  ?<;, 
g  =  md  +  ne  +  rj\  etc.,  will  give  the  same  values  to  m,  n,  and  r\  etc.,  etc. 

There  is  no  general  method  of  determining  that  a  series  is  absolutely  vot 
recurring.  The  best  practical  method  of  procedure  is  to  assume  jfird  that 
the  dependence  is  upon  tico  terms:  if  this  does  not  give  a  scale  which  will 
extend  the  series,  try  whether  the  dependence  is  not  upo'i  three  terms,  then 
upon  four,  etc.  Of  course,  applying  this  process  to  an  infinite  series  would 
not  determine  that  the  series  was  absolutely  not  recurring. 

Examples. 

1.  Find  the  scale  of  relation  in  the  series  1,  12,  48,  384,  1920,  etc. 

Suo's. — Assuming  that  the  dept^ndenco  is  upon  two  terms,  we  have  48  =  in 
+  12h,  and  384  =  \2m  +  A'Sti  ;  whence  m  =  24,  and  n  =  2.  Now  since  1J)20 
=  24-  48  4-  2   384,  we  conclude  that  +  24,  +  2,  is  the  scale. 

2.  Find  the  scale  of  relation  in  tlic  series  1,  O.r,  IS.c^  48.r\  120;/^ 
etc 

Suo's. — We  have  12.t'  =  m  +  Qxn,&nd  48^;'  =  (j.rm>  +  12.c*;t ;  whence  m—(ix*, 
iind  w=.T.  Now,  as  120c*  =  Gx*  ■  12x*  +  x-  48x*,  we  conclude  that  the  scale  of 
relation  is  +  6.c*,  +  t. 

3.  Find  the  scale  of  relation  in  the  series  1,  4.c,  C)x\  lla:*,  28:c*,  63.T*, 
and  extend  the  series  two  terms. 

Scale  of  relation,  +3ar',  —  .r^  +2.c 
Next  two  terms,  131.r*,  283.r'. 

4  to  11.  Find  the  scale  of  relation  in  the  following,  and  extend 
each  series  2  or  3  terms : 

(4.)  1,  X,  2x%  2x\  S:c*,  3.c\  4x\  Ax\  etc. 

(5.)  1,  3,  18,  54.  243,  720,  201G,  8748,  etc. 

(6.)  1,  X,  bx\  13:r\  41.^^  121.r',  d(jbx\  etc. 

(7.)  1,  4,  12,  32,  80,  etc. 

(8.)  3,  bx,  l7^,  \^T?,  23 A  45a;',  etc. 

,^.  a        ac     ac?    .,       a&   ,     . 


SERIES.  277 


(10.)  1,  4,  10,  20,  35,  56  84,  120,  etc. 
(11.)  1,  4,  8,  13,  10,  26,  34,  etc. 


814,  JProb,  2, — To  find  the  nth  tenn  of  a  series  when  a  svf- 
ficient  number  of  terms  is  given, 

Soi.UTiON. — Tlie  best  method  of  doing  this  depends  upon  the  chara<'ter  of 
the  series.     We  give  the  following  : 

1st.  Tlie  formula  I  z=:  a  -\-  {ii  —  \)d,  and  I  =  ar^-^,  resolve  the  problem  for 
arithmetical  and  geometrical  series,  I  being  any  tenn. 

2d.  Tlie  scale  of  relation  may  be  determined  by  Puou.  1,  and  the  series 
extended  to  the  nth  term  by  means  of  it. 

3d.  But  the  first  tenns  of  the  successive  orders  of  differences  afford  one  of 
the  most  elegant  and  general  methods.     Thus  from  {310)  we  have 

J)j——a+b;  .'.  b=a+I>i  ; 

2),—     n—2b+c;  /.  c=a+2D,+T)z  ;* 

i?3  =  -rH-;3Z»-3c+rf  ;  .-.  d=a+SDi  -hlWz+Di',  f 

Dj=     n+Ab-C)c+4d—c;  .'.  e=a+4n, +01)2+41) ,+1)^  ; 

i>,  =  -/H-;V>-10c+10ff-5«+/;  .*.  f=a+5Di  +10i>2  +  102>,,+5i)4  +  Z?5. 
etc.,             etc.,             etc. 

Whence,  by  induction,  we  have,  in  general,  the  72th  term  ~  a  +  {n  —  \)D, 

(n  -  IXw  -  2)  ^        (n  -  l)(7i  -  2){7i  -  3)  ^  .„    , 

4-  ^^ S ^2  +  ...      '  D3  +  etc.,  till  the  term  containing 

«  I  o 

D,_i  is  reached,  or  till  an  order  of  differences  is  reached  of  which  each  term 
is  0.  It  is  only  in  the  latter  case  that  the  method  is  practically  useful,  since  to 
determine  the  first  terms  of  the  n  —  1  successive  orders  of  differences,  requires 
that  n  terms  of  the  series  be  known. 


Examples. 

1  to  5.  Solve  the  following  by  means  of  the  scale  of  relation  : 

(1.)  Find  the  8th  term  of  1,  2x,  Sx\  28r',  100.9.^  etc. 

(2.)  Find  the  9th  term  of  1,  3«,  6x%  W,  W,  XW,  etc 

(3.)  Find  tlie  10th  term  of  1,  3.r,  2.r',  -  .r',  -  3.r*,  —  2r?;»,  etc. 

(4.)  Find  the  12th  term  of  3,  5,  7,  13,  23,  45,  etc. 

(5.)  Find  the  11th  term  of  1,  1,  5,  13,  41,  121,  etc. 

6  to  12.  Solve  the  following  by  means  of  the  successive  orders  of 
differences : 

(G.)  The  12th  term  of  1,  5,  15,  35,  70,  126,  etc. 


*  c  =  -  tt  +  2ft  +  2)3  =  -  a  f  2(a  4  X>,)  +  7)2  =  a  +  22),  -t  /),. 

t  d  =  a  -  3ft  f  ;ic  +  /),  =  o  -  3<a  )-  /),)  ♦-  3(a  +  %D^  +  D,)  f  />,  =*  a  ^  3/),   \  3/J,  t  />,. 


278  APPENDIX. 

(7.)  The  15th  term  of  1,  3,  6,  10,  15,  21,  etc.    Also  the  nth, 
(8.)  The  ?^th  term  of  1-2,  2-3,  3-4,  4-5,  etc. 
(9.)  The  12th  term  of  1,  4a:,  6x\  ll:r^,  28a;*,  GSx\  etc.* 
(10.)  Solve  the  first  five  given  above  by  this  method,  when  it 
will  apply.     Also  determine  the  scale  of  relation  in  (6)  to  (9)  in  cases 
in  which  the  series  is  recurring. 

(11.)  Find  the  wth  term  of  1,  2*,  3',  4^  etc. 

(12.)  Find  the  9th  term  of  70,  252,  594,  1144,  1950,  etc. 

13.  Extend  the  following  to  10  terms  by  the  metliod  of  differences*. 
1,  4,  8,  13,  19,  etc  Also  x%  4x\  Sx\  ISa^,  ldx'%  etc.  Also  1,  6,  20, 
50, 105, 196,  etc, 


SIS,  JProb,  3» — To  determine  whether  a  series  is  convergent  or 
divergent. 

Solution. — 1st.  When  theternu  are  nil  -\- .  If  the  series  is  not  decreasing, 
of  course  it  cannot  be  convergent.  Thus  a^-b  +  c  +  d  +  e  +  etc.,  \i  n  <h 
<  c  <  d  <  e,  etc. ,  is  >  a  oo  .  I^t  us  then  consider  the  case  wlieu  the  terms  are 
all  + ,  and  a  >  b  >  c>  d>  e,  etc.     We  have 

S=a  +  b  +  c  +  d  +  e  +  etc.  =  a(l+-+  -+-  +  -  +  etc. ) 

\        a      a      a      a  / 

(b      cb      deb      edcb  \ 

l  +  -  +  i-+^-+  :r-r  +  etc. ). 
a      ba      cba      dcba  / 

Now  if  -,     -,     -,     -,  etc.<.p,    S <  n{\  +  p  +  p^  +  p'^  +  p*  +  etc.),  which,  if 
a      b      G      a 

;><!,= .    Therefore,  An  injinite  series  of  positive  terms  is  always  conter- 

1  —p 

gent,  if  tlie  ratio  of  each  term  to  the  preceding  term  is  less  than  some  assignable 

quantity  which  is  itself  less  than  1. 

2d.  When  the  terms  are  alternately  f  and  — ,  and  decreasing.  Let  the  series 
he  a,  —b,   +  c,  —  d,   +  <",  —  etc.     Now  we  may  write 

5  =  («  —  Z»)  +  (c  —  rf)  +  («  -/)  +  etc. ; 
and  also  8=  a —  {b  —  e)  —  {d  — e)  —  etc. 

Since  the  terms  are  decreasing  {c  —  d),  («—/),  «tc.,  are  +,  and  S>a  —  b. 
Again,  (6  —  c),  {d  —  e),  etc.,  are  + ,  and  8  <  a.  Therefore,  Any  series  of  decreas- 
ing terms,  which  ttrms  are  alternately  +  and  — ,  w  convergent. 

3d.  When  the  terms  are  alternatdy  +  and  — ,  and  increasing,  we  have 
8=ia  —  b-\-c  —  d  +  e  —f  +  g  —  etc.  =  a  —  {b  — c)  —  {d  — e)— {f  — g)  —  etc. 
Now,  since  the  terms  are  increasing,  b  —  c,  d  —  e,  f  —  g,  etc.,  are  essentially 
negative.     Representing  these  differences  by    —  d,  —d^,   —d^,  etc.,  we  have 


*  It  is  evident  that  the  12th  term  involves  x  to  the  11th  power,  or  contains  a:".     I.cncc  we 
have  only  to  find  the  coefficient,  or  the  13th  term  of  the  scries  1,  4,  6, 11,  28,  63,  etc. 


SERIES.  279 

S  =  a  +  d  +  d^  +  di  +  etc.,  a  series  which  can  be  examined  by  the  first  process 
given  above, 

4th.  The  process  of  grouping  the  terms  and  thus  forming  a  new  series,  as  in 
the  last  case,  is  frequently  serviceable  in  other  cases  than  that  there  specified.* 


Examples. 

1.  Determine  whether  1  -f  -  +  — -  +  _-_  +  +  etc.,  is 
a  convergent  series. 

Sug's. — Here  -  =  1,     t  =  ^,      -  =  o>     -t=  t>  etc. :  whence  we  see  that  each 
a  b      2       c      d      d      4: 

of  the  ratios  after  the  second  is  less  than  ^,  which  is  itself  less  than  1.     Hence 

the  series  is  converging. 

2.  Determine  whether  l+J  +  ^  +  j^-  etc.,  is  a  converging  series. 
3  to  6.  Determine  which  of  the  following  are  converging : 

(3.)  i  +  i  +  i  +  iV  +  etc. 

(4.)  1  +  -  +  — 2  +  -3  +  etc.,  r  being  >  1,  i.  e.,  any  decreasing 
geometrical  progression. 

.    .  _3_  4  5 

^  ^^  1.2.2  "^  2.3.2'^  "^  3:4:2^  ^  ^*^- 

2^'  iC^  (K^  S^  3^ 

7.  For  what  values  of  a;  is   x r  +  —  —  —  -f-^ 4-  etc., 

2        3        4        5         6 
convergent,  and  for  what  divergent  ? 

Bug's.— For  a;  ~  1  we  have  a  series  with  the  terms  alternately  4-  and  —, 
and  decreasing.     Hence,  by   (t/iJ,  2d),   the   series   is  convergent,     /gain,  to 

examine  the  series  for      «>  1,     it  may  be   written      x  — —  +  x^\- ~) 

2  \  8        4  / 

+  a^  |— —I  +  x^  I— —j  +   etc.        Now,   for    ic  >  1,     some  one  of  the 

factors     (-Q-~x)'        \6~~q)'        (t"^)'     ^*^"'       ^"^  ^^^   following  it 

8  \      X 

will  become  negative.    Thus,  if  a?  =  ^,  all  following  ^  —  5   will  be  negative. 

7  7      8 

*  This  is  confessedly  quite  an  imperfect  presentation  of  this  prohlem ;  but  it  is  eufficient  for 
most  purposes,  and  i»  as  fUIl  as  our  limits  will  allow. 


280  APFKNDTX. 

The  sum  of  that  portion  of  the  series  preceding  this  first  negative  factor  will  be 
finite,  since  it  will  be  composed  of  a  finite  number  of  finite  terms.  Let  us  now 
examine  the  infinite  series  which  is  composed  of  negative  terms.  Let  a  be 
the  value  of  x  for  which  we  are  examining  the  series,  and  y  the  exponent  of  x 

in  the  first  negative  term.     This  term  is  therefore  (ffy \ .     Now  this 

may  be  taken  as  the  general  term  of  this  portion  of  the  series  if  we  understand 
that  a  is  constant  and  y  variable.     As  y  increases  by  2  in  each  successive  term, 

the  first  two  terms  of  this  series  are  api ;  ) ,    «"+'(  r  •—         .,  |  ;  and 

the   ratio  of  the  second  to  the  first   is  a^  \l±lr^y:Z^    x     J^^-^lJ- 

(  (y  +  2) (y  +  3)  y  +  \-(iyS 

—  ^    ]  7V^ — r-7^    /o      ..  ^    *       /i/ — n  ^         n [  >  the  limit  of  which,  as  y  in- 

creases  to  infinity,  is  a*.  But  as  /7  >  1 ,  n^  >  1,  and  this  negative  series  is  diver- 
gent and  its  sum  is  infinite.  Hence  the  given  series  is  convergent  for  «  ~  1,  and 
for  all  values  of  a;  >  1  it  is  divergent. 


316»  JProb,  4. —  To  find  the  sum  of  n  terms  of  a  series. 

This  problem,  like  many  others  concerning  series,  does  not  admit  of  a  general 
solution.     We  specify  the  following  cases  : 


ar  convergent,  for  n  finite,  8=in[2a  +  (n  —  l) d],    or    S  =        __  —  .     For  an 


Case  1. —  W^ien  the  series  is  Arithmetical  or  Geometrical,  either  divergent 
convergent,  for  n  finite,  8  =  in[2a  +  {n  —  1) d] 

infinite  geometrical  convergent  series  we  have   S  =  7~~'  • 

Case  2.—  When  the  series  is  an  infinite,  decreasing,  recurring  series,  to  find 
the  sum  of  the  series  (t.  e.,  n  being  oo).  Let  the  series  be  a  +  b  +  c-¥d  +  e+  etc., 
and  m,  n  the  scale  of  relation,  the  dependence  being  upon  two  terms.  Whence 
we  have 

a  =  a, 

b  =b, 

c  =  am  +  bn, 

d  =:  bm  +  en, 

e  =  ofn  +  dn, 

f  =  dm  +  en, 


Putting  S=a  +  b  +  e  +  d^ etc., 


and  adding,  this  gives      8=a  +  b  +  8m  +  {8—  a)n. 

Solving  for  8,  we  hava  8  =  ^'^^"'^ .  (1) 


SERIES.  2S1 

When  the  scale  of  relation  consists  of  three  terms,  as  m,  n,  r,  '.vo  i:av« 

a  :=  n, 

b  =b, 

c  =  c, 

d  =  am  +  bn  +  cr, 

e  =  bm  +  en  +  dr, 

f  z=  cm   +  dn  +  er, 

g  =  dm  +  en  +  fr. 


Whence  8=  a  ^-b  -v  c  -v  Sm  +  {S  —  a)n  +  {S  —  a  —  b)r. 

.,,..„  ^      a  +  b  +  c  —  an  —  {a  +  b)r 

And  solving  for  S,         S  = :; —  .  (2^ 

1 —m  —  n  —  r  ^ ' 

When  the  scale  of  relation  consists  of  four  terms,  as  m,  n,  r,  »,  we  can  write 

from  analogy, 

n  ->rb->r  e  ■¥  d  —  an  —  {a  +  byr  —  {a  ^-b  +  e)8  ,_. 

^  — z .  (o) 

l—m—n—r—s 

Case  3. — To  find  the  sum  of  n  terms  of  a  series  by  the  method  of  differ- 
ences.— Let  the  series  be  a,  b,  c,  d,  e,f,  etc.,  which  we  will  call  {A). 
Now  if  we  write  the  series 

(B)       0,a,a+b,   a  +  b  +  c,   a  +  b  +  e  +  d,   a  +  b  +  c+d  +  e,  etc., 

of  which  the  scries  (A)  is  the  first  order  of  differences,  it  is  evident  that  the 
(71  +  l)th  term  of  (B)  is  the  sum  of  71  terms  of  the  given  series  (.4).  By  tho 
formula  for  the  nth  term  {314:,  3d),  which  is 

The  nth  term  =  a  +  („-l)Z),  +  f!^!^  D,  +  („-lX»-2)(n-3)^^^  ^^  ^ 

A  jo 

noticing  that  a,  the  first  term,  in  series  {B)  is  0,  that  D^  of  series  (J5)  is  a  of 
series  {A),  D^  of  series  {B)  is  Di  of  series  (-4),  etc.,  we  have,  for  the  sum  oi 
n  terms  of  (A) 

n(n  -  1)  _.        n{n  —  Vj{n  -  2)  ^ 
S=na  +  -^-- — -'  Dy  +  -5^ 1^^ '  I>i+  etc. 

On  this  formula  we  observe  that  when  the  orders  of  differences  do  not  vanish, 
if  the  series  is  extended  to  the  (;i  +  l)th  term  the  coeflBcient  of  that  term  will 
become  0,  and  the  series  will  terminate. 

Moreover,  in  cases  in  which  the  nth  order  of  differences  vanishes,  the  same 
number  of  terms  of  this  formula  will  give  the  sum  of  any  number  of  terms  of 
the  series  above  the  nth. 

Case  4. — Upon    the    principle    that    any    fraction  of    the    form    — — ^ — c 

n{n+p) 

=  -  ( ~ —  )  ,*  many   series   of   fractional   terms   of  the   form   — — - — r 

p\n      n  +  p/  n{n  +  p] 

may  be  summed. 

*  This  is  evident  since  ^  -  -^  =  ^^^  "  "^  =  -^l~. 
n       n+p  nin-i-if)  n{n  +  p) 


^Bj^  appendix. 

Also  many  series  of  fractional  terms  of  the  form   — f r— ;  may  be 

^  n{n  +  p)(n  +  2p) 

summed  from  the  fact  that 


n{n  +  p){n  +  2p) 


=  ii_5 ? I. 

2p  ( n{n  +  p)      (n  +  p){n  +  2p) ) 


When  the  fractional  terms   are  of  the  form — ^  ^  .^ -—.,  the 

n{n  +  p){n  +  2p){n  +  dp) 


summation  may  often  be  effected  upon  the  principle  that 
9  _    1    i  9 


—1 L 

in  +  2o)(n  +  3») ) 


n{n  +  ^X^  +  2;))(w  4-  3p)        Sp  ( 7i{n  +  p){n  +  2p)       (n  +  p){n  +  2p\n  +  '6p) 

The  practicability  of  this  method  depends  upon  our  ability  to  find  the  differ- 
ence between  two  series.     Tims,  when  the  terms  of  the  given  series  are  of  the 

form     ,     — r- ,  if  we  can  find  the  difference  between  two  series  whose  terms 
n{n  +  p) 

are  of  the  form  -  ,   and  — - —  respectively,  we  can  find  the  sum  of  the  given 

n  n  ■¥  p  ^ 

series.     But  the  method  will   be   more   readily  comprehended   in  connection 
with  its  application.     (See  Ex's  15-30, ) 

Examples. 
1  to  7.  Find  the  sum  of  the  following  recurring  series : 

(1.)  1  +  2a;  +  8a:»  +  28x^  -f  lOOx^  +  etc. 

(2.)  1  +  2a:  +  3ar  4-  Sa:"  4-  8ar*  4-  etc. 

(3.)  1  +  3^;  +  5ar«  -H  T-c*  +  etc. 

(4.)  3  4-  5a;  +  7a:'  +  133:^  +  23a:<  -f-  45a:5  +  etc. 

(5,)  1  +  1  +  5  4-  13  +  41  +  121  4-  etc. 

(6.)  1  4-  a:  +  2a;»  4-  22^*  4-  32:*  4-  32:*  4-  4.c'  4-  4a:'  4-  etc. 

,^.  a      ac         a(?  ,      ^'   , 

(7.)  j-jr^+-jr^--jr^  +  etc. 


8  to  14.  Find  the  sum  of  the  following  by  the  method  of  differences : 

(8.)  1  4- 3  4- 5  4- 7  4- etc.,  to  20  terms  ;  to  n  terms. 

(9.)  14-24-3 +  44-5 +  etc.,  to  50  terms;  to  n  terms. 
(10.)  14- 5 4- 15 4- 35 4- 70 4- 126  + etc.,  to  30  terms;  to  w  terms. 
(11.)  70  +  252  +  594  +  1144  +  1 950 etc., to 25 terms; to ^ terms. 
(12.)  1  +  2* +  3* +  4*  + etc.,  to  12  terms;  to  n  terms. 
(13.)  1  +  2'  +  3'  +  4'  +  etc. ,  to  n  terms. 
(14.)  1  +  2' +  3' +  43  + etc.,  to  n  terms. 


15.  Find  the  sum  of  — ^  +  ^r-r  +  r-j  +   — ^  +  etc.,  by  the  method 

x'tii        Z'O        0''±        4'0 

given  in  Case  4. 


SERIES.  283 

Sug's. — If  we  put  p  =  l,  q  =  l,  and  n  =  1,  2,  3,  4,  etc.,  successively,  the 

general  form  of  the  term  in  this  series  is     ,      — .     Thus  we  have 

n{n+p) 

For  the  Ut  term.    ^_  =  j^_  =  -^  (j  -  ^1--)*=  1  (l  -  ^  ) 

For  the  2d  term.    ^^^^  =  ^-^-  =  ^  (1  -  ^-3)*=  1  (i  -  1  ) 

For  the  3d  term.    ^-^  =  __L_  =  ^  (^  _  ^).=  1  (^  _  1 ) 

For  the  4th  term,   -— ^^^  =  ,,/    ,,  =  1(1-  7"^-?)*=  ^  (^7  -  ^  ) 
w(;i  +  p)      4(4  +  1)      1  \4      4  +  1/  \4      5  / 

etc. ,         etc. ,         etc. 
Putting  S  for  the  sum  of  the  series  and  adding,  we  have 

_  (l+i  +  i  +  i  +  etc. )  _ 
~i     -i-^-i-etcf  ~  ^• 

Note, — It  will  be  seen  that  this  method  is  only  an  ingenious  device  for  de- 
composing the  given  infinite  series  into  two  infinite  series,  one  of  which  destroys 
all  but  a  finite  portion  of  the  other. 

16.  Find  the  sum  of  — x  +  r-r  +  ;=-^  +  l^r-^  +  ^^c- 

l«o       O'D       0*7       7*y 

17.  Find  the  sum  of  7i  terms  of  each  of  the  two  preceding  series. 
SuG. — We  have  for  the  7ith  term  of  the  last  series  q  =  l,    p  =  2,    7h  =  2n  —  l, 

since    2n  —  1    is   the  nth  odd   number.      Hence   for  the  nth   term 

=  77 1  s T  —  s 7  )  •      ^^  therefore  have 

3  \2n  —  1       271+1/ 


-^ 


.111  1 

1  +  ^   4-  -;r  +  ^ 


3       5    ■    7  2n-l 

111  11 


8       6        7  2n  -  1      2n  +  1 

_       ^ 

~  2n  +  1' 


2\         2n  +  lJ 


18.  Find  the  sum  of— 7  +  --^  +  7— ;  +  — -+  etc.  Also  of  n  terms 

1'4      2'0       d-6       4-7 

of  the  same. 

19.  Find  the  sum  of— r  —  773  +  — — — -  +  etc.,  to  n  terms. 

15       60       63      99 


Since  by  Case  4,    -^^—  =l.(t-  -^'^-). 
n{n  +  p)      p  \,i       n  .  pj 


284  APPENDIX. 

SuG's.— It  will  be  seen  that  this  series  is  the  same  as  r—  —  — -  +  =-r-  —  ^—rr 

+  etc.     Hence  by  making  q=2,S,  4,  5,  etc.,  successively,  n  =  S,  6,  7,  9,  etc., 
successively,  and  p  =  2,vre  have 

ii(5  -  f)  -  a  -  f)  +  (f  -  I)  -  (f  -  A)  +  etc.},     or 
i!l  -  a  +  f)  +  (f  +  ^)  -  (^  +  t)  +  W  +  etc.} 
=  ili-l  +  l-l+-A-  +  etc.}. 

Now  the  form  of  this  last  term  is -r  ;   and  if  an  even  number  of  terms  of 

2n  +  6 

1(2  n  +  1  ) 

the  criven  series  is  taken,  we  have  ;r  ■{  t  —  1  +  s r^  c  .  ^^^  *l^e  intermediate 

^  2  id  2n  +  6) 

terms  destroying  each   other.       But   if    an  odd  number  is  taken,   we   have 

l/2n  +  l\^.     „  n  +  1        1  1  ,  . 

— (  —  — ).     Fmallv,  as  ^ =  -^  —  ^-7^7 77, ,    we  have  for  an  even 

2\  3      2/1  +  3/  •  2/1  +  3       2      2(2/i  +  3) 

number  of  terms i  ||  -  [  -  ^^^^ [  >  <-  ^  -  J^"?,  =  -<»  for  an  odd 

number.  |||  -  ^  +  2^-3,}  -    <>'  jV  Sca^T^T)"    '"''''^  "=*•'"'  ''*'" 
=  0 ;  whence  the  sum  is 


4(2/1  +  3)  12 

20.  Find  the  sum  of  :j-^  +  — r  +  5—  +  etc 

1*0  Z'^  iJ'O 

21.  Find  the  sum  of  ^r-s  —  -t-t  +  jr-;^  —  etc. 

I'O  Z-i         O'O 

22.  Find  tlie  sum  of  -—  +   — —  +  — —  +  etc. 

O'O  D'l>i  y«l0 

8uo.-Tbi8  equals  i(j-L  +  _!-  +  _L  +  etc.). 

23.  Find  the  8um  of  ji  +  ^  +  ^  +  jgi-^  +  etc. 

24.  Find  the  sum  of  — — -  +  ^r-5-7  +  5-7-^  +  etc. 

I'ii'O  4-6-4:  6'4:-0 

Sdg's. — By  putting  p  =  1,  q  =  4,  5,  6,  etc. ,  successively,  and  71  =  1 ,  2,  3,  etc., 

successively,    these     terms    take    the     form      —7 ^7 7.-^, ,      and    since 

71  {n  +  p)  {n  +  2p) 

-^ ^-—  =  ;r-  ^ ^ — -,  V  .  we  may  write  the  given 

//(/I  +  p) {)i  +  2p)      2p  in{n  +  p)      (w  +  p){n  +  2p)) 

series  thus : 

1 1  (ri  -  2^3)  +  (2^3  -  3^4)  +  (3-4  -  4-5)  +  '=*^-  \ 


PILING   BALLS   AND   SHELLS. 


m 


4  5  6 


2   3         3-4 


etc. 


=  l(^  +  2^  +  5^4  +  «*"•)  -  1(2  +  i)  (^-  E'^- 1^  =  li- 


25.  Find  the  sum  of 


15 


5.8.11       8-11.14  '   11.14.17 


+  etc. 


36.  Find  the  sum  of  ^2^^  +  ^^  +  _A-  +  etc. 

1  4  7  10 

27.  Find  the  sum  of  ^^  +  ^^^  +  .-^^  +  ^:^^  +  etc. 

28.  Find  the  sum  of  ^-:^.t-.  +  :rJ^—  +  .-r-rV-r  +  etc. 


1.2.3.4       2.3.4.5   '   3.4.5. G 


SuG. — Consider      that 


I 


? I 

{n  +  p)  {n  +  2p)  {n  +  'Sp)  f 


n  (n  +  p)  {11  +  2p)  {n  +  3^)  'dp  ( n  {n  +  p)(n  +  2p) 


29.  Find  the  sum  of 


+  - 


1.3.5.7       3.5.7.9       5.7.9.11 


-f  etc. 


30.  Find  the  sum  of 


r^  + 


3.6.9.12       6-9.12.15       y. 12. 15-18 


+  etc. 


Note. — The  above  examples  are  taken  from  Young's  Algebra,  an  excellent 
»>ld  Englisli  work  to  which  American  editors  are  much  indebted. 


Piling  Balls  an^d  Shells. 

317*  In  arsenals  and  navy-yards,  cannon-balls  and  shells  are 
piled  on  a  level  surface  in  neat  and  orderly  piles  of  three  different 
forms,  viz.,  triangular,  square,  and  ohlong.  The  figures  below  will 
sufficiently  illustrate  these  forms : 


_?  ^^  (2)®  {?!)"<§  <S)  (D  @  #' ®  t)  ® »; 

a  ^'  ®  #  ®  #®  ®  ®  <®  ®.  ®  #  ®  1 
^■(i)®.®?i)#®j)®<®<     

*)  (fS)  (S  •'1^  ®  ®  B>fiM'&. 


OBItOKG  PILB. 


TRIANGULAR  PILE.  SQUARE  PILK. 


318,  I^rop. —  The  formula  for  the  number  of  halls  or  shells  in  a 
triangular  pile  having  n  balls  or  shells  on  a  side  of  its  loioest 
Jn  (n  +  1)  (n  +  2). 


course  IS 


286  APPENDIX. 

Dem. — The  student  will  be  able  to  discover  that,  beginning  at  the  top,  the 
number  of  balls  or  sheila  in  each  course  is  as  follows  : 

],        1+2,        1+2  +  3,        1  +  2  +  3  +  4, etc., 

or  1,        3,        0,        10,        15,        21, etc. 

Summing  this  series  to  n  terms  by  the  method  of  differences  he  will  obtain  the 
formula. 

3 If),  Cor. —  The  number  of  courses  in  a  triangular  pile  is  equal 
to  the  number  of  balls  or  shells  in  one  side  of  the  lowest  course  ;  and 
the  number  of  balls  or  shells  i7i  the  lowest  course  is  1  +  2  +  3-1-4 
n,  or  J(n'  +  n). 

320,  Prop, — The  formxda  for  the  number  of  balls  or  shells  in 
a  square  j^Hc  having  n  balls  or  shells  on  a  side  of  its  lowest  course  is 

|n(n  +  l)(2n  +  l). 

The  student  should  be  able  to  demonstrate  tliis  as  above. 

321,  Cor. —  7'he  number  of  courses  in  a  square  pile  is  equal  to  the 
number  (f  balls  or  shells  in  one  side  of  the  lowest  course ;  and  the 
number  of  balls  or  shells  in  the  lowest  course  is  1  +  3  +  5  +  7  +  9 
2n  —  1,  or  n*. 


322,  Prop, —  Tfie  formula  for  the  number  of  balls  or  shells  in 
an  oblmig  jnle  having  m  balls  or  shells  in  the  length  of  the  base  and 
n  in  the  width  is 

in(n  +  1)  (3m -11  +  1). 

Dem. — Observe  that  there  are  as  many  courses  as  there  are  balls  in  the  width 
of  the  base.  Let  m  be  the  number  in  the  top  row,  whence  we  have  for  the 
number  in  the  successive  rows  from  the  top  downward, 

w',  2<w'  +-  1),  3(m'  +  2),  4(m'  +  3),  5(m'  +  4),  etc. 
Taking  the  successive  differences,  we  find  i),  =  ?»'  +  2,  Dg  =  2,  and  Dg  =:  0. 
Substituting  in 

8=^^ 2 ^^^ 2T3 ^*' 

nin  —  1) ,    ,       -,      n{n  —  l)(n  —  2)       ...  ... 

we  have  S  =  m'n  -+  -^^-^ — -  (m'  +-  2)  H ,  which  readily 

«  o 

reduces  to  8 -  \n\{n -^  \) {^m' -\- 2n  —  2)} . 

Now  m  being  the  number  of  balls  or  shells  in  the  length  of  the  base,  wo  observe 

that  m!  ■=m  —  n+\,  which  substituted  in  the  previous  equation  gives 

ScH. — If  we  make  m  =  n,  this  gives  the  formula  for  the  square  pile,  as  it 
should. 


EEVERSION   OF   SERIES.  287 

Examples. 

1.  Find  the  number  of  balls  in  a  triangular  pile  of  20  courses. 
In  a  triangular  pile  with  42  balls  on  one  side  of  the  lowest  course. 
How  many  balls  in  the  bottom  course  ?  How  many  in  one  of  the 
faces  ? 

2.  Find  the  number  of  shells  in  a  square  pile  with  30  courses. 
With  23  balls  in  one  side  of  the  lowest  course.  With  2209  in  the 
bottom  course.    How  many  balls  in  one  face  of  each  pile  ? 

3.  Find  the  number  of  balls  in  an  oblong  pile  whose  bottom 
course  is  42  balls  by  20.  Whose  top  course  contains  23  balls,  and 
which  has  fifteen  courses. 

4.  How  many  shells  remain  in  an  incomplete  triangular  pile  whose 
top  course  contains  28  shells,  and  whose  bottom  course  has  15  shells 
on  a  side  ? 

5.  How  many  balls  in  an  incomplete  square  pile  wliose  top  course 
is  8  balls  on  a  side,  and  whose  bottom  course  is  20  balls  on  a  side  ? 

8.  How  many  shells  in  an  incomplete  oblong  pile  whose  top 
course  is  12  by  20,  and  whose  bottom  course  is  62  shells  in  length  ? 


Reversion-  of  Series. 

323.  Prob, —  To  revert  a  /Series. 

Solution. — The  problem  is,  having  given 

/(y)  =  ax+bx*  -^  cx^  +  dx*-  +  etc.,  {A) 

to  express  a;  as  a  function  of  y,  i.  e.,  to  obtain 

x  =  Ay  +  By^  +  Cy*  +  Dy*  +  etc.,  (B) 

the  essential  thing  in  the  solution  being  to  find  the  values  of  the  indeterminate 
coefficients  A,  B,  C,  D,  etc.  To  do  this,  we  form  x^ ,  x^,  x*,  etc.,  from  (B)  in 
terms  of  y,  and  substitute  in  the  second  member  of  (A).  Whence  we  have 
f(y)  =/'  (y)*  From  this  relation  we  can  obtain  the  values  of  the  indeterminates 
A,  B,  C,  D,  etc.,  in  the  ordinary  way. 

Examples. 

1.  Given  y  =  x  -\-  ^x^  +  \oi^  +  ^x"^  ■{■  etc.,  to  revert  the  series,  i.  e., 
to  express  the  value  of  a;  in  a  series  involving  y. 


*  This  notation  mean?  that  both  members  are  functions  of  y,  bat  that  they  are  not  the  same 
function :  one  is  the  /function,  and  the  other  the  /'  function. 


288 


APPENDIX. 


SuGs.— Assume      x  =  Ay  +  By*  +  Cy^  +  By*  +  etc. 
Whence  x'  =A*y*  +  2ABy^  +  2AC    I  y*  +  etc., 

+     i?M 

and  X*  =  A*y*  +  etc.,  these  developments  being  extended 

as   far  as   is   necessary   in  order    to   determine   four    terms   of    the    reverted 
eeries. 

Substituting  these  values  in  the  given  series  we  have 


y  =  Ay  +    B      y*  +      C 

y'  +     D 

y*  +  etc. 

+  M*          +  AB 

+   AG 

+  U' 

+    \B' 

+A'B 

-4-     U* 

Whence  ^  =  1,  B  +  iA*=0,C  +  AB-\-^A'  =  0,  and  B+AC+^B'+A'B 

+  iA*=0.     These  give  ^  =  1,  B=  -^  C=^,  and  i?  =  -  -/,-.     Therefore 

the  reverted  series  is          ^  =  y— ■Ky'+-^ 

i            [9 

y^-iy^  +  eta 

2  to  6.  Revert  the  following : 
(2.)  y  =  X  +  x'  +  x"  +  a^  +  etc. 
(3.)  yzzzx  +  Sx'  +  bxf'-^lx'-hdr^-h  etc. 
(4.)  i/  =  x-ijr'-\-ix^~\x'  +  etc.* 
(5.)  i/  =  2x  +  Sx"  -^  4a^  -^  6.r'  +  etc. 
(6.)  y  =  1  +  a:  +  K  +  ,K  +  i^  +  etc.f 

7.  Required  to  express  the  value  of  y  in  tei'ms  of  x  from  the 
relation 

y  •\-  ay^  ■\-  by^  -\-  cy*  +  etc.  =  mx  +  ut^  -\-  px^  +  qx^  -\-  etc. 


Interpolation. 

S24.  Prob, — Having  given  a  series  of  functions  a,  b,  c,  d,  e, 
etc.,  to  find  a  function  intermediate  between  any  two  of  this  series, 
which  function  shall  conform  to  the  laio  of  the  series. 

III.— Let  the  series  of  functions  be  the  logarithms  of  232,233,  234,  235,  etc., 
viz.,  2.365488,  2.367356,  2.339216,  and  2.371068  ;  let  it  be  required  to  find  the 
logarithm  of  233.4,  i.  e.,  the  function  I  of  the  way  from  2.367356  to  2.369216. 

Solution. — The  solution  of  this  problem  is  simply  an  application  of  the 

*  In  this  example  it  will  be  more  expeditioas  to  assume  xz=  Ay-\-  Bjfi  +  Cy^  ^  etc  ,  though  it 
is  not  essential. 

t  Tranepose  the  1,  put  z.=  y-1,  and  then  revert  the  series  z  =  x  +  ^t^  +  .\-xi+.L.x*  +  iiic. 

This  i^  necessary,  since  the  theory  of  Indeterminate  Coefficients  assumes  that  both  variables 
hecome  0  at  the  same  time  ;  i.  «.,  that  z=0,  makes  z=Q, 


INTERPOLATION  OF  SERIES. 


289 


formula  for  finding  the  nth  term  of  a  series  by  the  Method  of  Differences 
{:il4);  viz., 

The  fith  term  =  a  +  {n  —  l)i>i  +  ^^ ^ B^  4-  ^^ ,  ^ •  i>3+etc. 

But  for  our  present  purpose  it  is  more  convenient  to  replace  the  {n—1)  of  the 

formula,  where  n  represents  the  number  of  the  term  sought,  by  -  ,  a  fraction 

which  indicates  the  distance  of  the  term  sought,  from  the  first  term  used, 
this  distance  being  measured  by  calling  the  distance  between  any  two  given 
terms  1.    Thus  in  the  series  a,  b,  c,  d,  e,  etc.,  a  term  i^  of  the  way  from  &  to  c, 

would  be  reckoned  at  a  distance  \%,  or  '3  from  a,  i.  c,  -  would  be  §  in  this  case. 

<l 
Now  by  this  method  of  reckoning  it  is  evident  that  the  {n—1)  of  the  formula  must 

be  replaced  by  -  ,  for  w  stands  for  the  number  of  the  term,  which  is  one  more ' 

than  the  number  of  intervals  between  it  and  the  first  term.     Thus  the  4th 

P 
term  is  3  intervals  from  the  first  term.     Making  this  substitution  of  -   for  w  — 1 

the  formula  becomes 


,+etc. 


Termtobeinterpolated=a+  ^i>,  +  ^^(^-1\D2+^  ^(^  -l)  (^-2\d 

325.  ScH.  1. — On  this  formula  we  observe  that  when  the  series  of  func- 
tions is  such  that  the  differences  vanish,  i.  e.,  D^,  D^,  D^,  or  some  order 
becomes  0,  the  formula  gives  an  absolutely  correct  result.  But  when  the 
differences  do  not  vanish,  the  result  is  only  an  approximation.  However, 
such  is  the  closeness  of  approximation,  that  for  practical  purposes  only 
second  differences  are  usually  needed,  although  sometimes  third  and  fourth 
become  necessary. 

Examples. 

1.  Finding  from  the  tables  the  logarithms  of  232,  233,  234,  235, 
to  be  2.365488,  2.367356,  2.369216,  and  2.371068,  required  to  inter- 
polate the  logarithm  of  233.4. 

SOLUTION. 


ARGUMENTS.* 

FUNCTIONS. 

1st  diff's. 

2d  diff's. 

3d  diff's. 
.000000 

232 

233 
234 
235 

2.365488 
2.367356 
2.369216 
2.371068 

.001868 
.001860 
.001852 

-.000008 
-.000008 

*  In  such  acas«  the  number  is  called  the  -4 rflrt»we»e,  a^4  \X»  logaritl^m  Ihp  A^nptipn.    This 
means  simply  that  the  logarithm  is  f^  fuuctjon  of  ^lie  number  (or  argumppl). 


290 


APPENDIX. 


In  this  case  a  =  2.3G5488,  Bi  =  .001808,  B^  =  -.000008,  D^  =  0,  and  ^  =  p 

q       5 

Ilcnce  we  have 

log  232  =  a     =  2.365488 

^  i>,  =       I  (.001868)  =       .002615 
q  5 

.'.  log  233.4  = 
Mctly  as  it  is  in  the  tables. 


2.368101,     which  is  ex- 


2.  Finding  from  the  table  the  logarithms  of  61,  62,  etc.,  interpo- 
late the  logarithm  of  62.23. 

S26,  ScH.  2. — When  second  differences  only  are  to  be  used,  and  four 

functions  of  the  series  are  known,  a  convenient  and  excellent  formula  is 

ol>tainecl  thus:  Let  the  four  functions  be  a,  6,  c,  d^  and  let  it  be  required  to 

v' 
interpolate  between  6  and  c.  Let  —,  be  the  interval  from  h  to  the  place  of  the 

term  to  be  interjiolated.  Now  if  we  compute  from  ft,  instead  of  from  a,  the 
])receding  formula  will  become 

The  interpolated  function  =b  +  ^\  Di  +2W~^)^M' 

in  which  Z),  is  the  second  of  the  first  differences,  i.  «.,  the  one  which  falls 
between  b  and  c ;  or,  in  general,  if  we  tabulate  the  differences  as  above,  it 
is  the  first  difference  which  falls  in  the  same  horizontal  line  with  the  func- 
tion to  be  interpolated.  Again,  as  the  second  differences  are  supposed  to  be 
different,  it  is  best  to  take  the  arithmetical  mean  of  the  two,  which  mean 
will  also  fall  in  the  same  horizontal  line  with  the  interpolated  function. 

3.  Find  by  (326)  the  logarithm  of  68.53  from  the  logarithms  of 
67,  68,  69,  70.    (See  table.) 


ARGUMENTS. 

FXJNCTIONS. 

IST   DIFF'S. 

2d  diff'b. 

MEAN  OF 

2d  diff's. 

67 

68 

* 

1.826075 
1.832509 

.006434 
.006340 
.006249 

-.000090 
-.000091 

-  .0000905 

69 
70 

1.838849 
1.845098 

Z),  =  .006340,    and 


Here    we    have    &  =  log  68  =  1.833509,  ^  =   ^ 

Df  =  —  .0000905.     The  student   should  make  the  substitutions  and  compare 
with  the  table. 


INTEUPOLATION   OF   SERIES. 


291 


327 -  ScH.  3.  —  But  it  is  not  for  interpolating  logarithms  that  this 
method  is  chiefly  used.  For  this  purpose  the  method  given  in  {196)  is 
preferable.  The  student  will  readily  discover  that  the  method  of  {106} 
is  identical  with  that  just  given  if  only  first  differences  are  used.  When 
great  accuracy  is  required,  and  the  tables  used  give  the  logarithms  to  8  or 
10  places,  it  sometimes  becomes  necessary  to  use  mean  second  differences, 
as  above.  It  is,  however,  in  Astronomy  that  Interpolation  has  its  most  im- 
portant applications.  Thus,  suppose  the  Right  Ascension  (analogous  to 
terrestrial  longitude)  of  a  planet  has  been  observed  four  times  at  intervals  of, 
say  one  day.  By  interpolation  we  may  find  its  Right  Ascension  at  each  interme- 
diate hour,  or  point  of  time.  In  this  problem  the  Right  Ascension  is  t^e  function, 
and  the  time  ia  the  argument. 

4.  Tlie  Right  Ascension  of  Jupiter  to-day,  July  1st,  at  noon,  is 
10h.5m.  38.6s.;  July  2d,  at  noon,  it  will  be  lOh.  6m.  18.86s.;  on  July 
3d,  lOh.  6m.  59.41s.,  and  July  4th,  lOh.  7m.  40.24s.  What  will  it  be 
July  2d,  at  midnight  ? 

SOLUTION. 


ARGUMENTS.* 

FUNCTIONS.* 

1st  diff's. 

2d  diff's. 

MEAN 

2d  diff's. 

July  1. 
July  2. 
July  3. 
July  4. 

lOh.  5  m.  38.6?. 
lOh.  6  m.  18.86  s. 
lOh.  6  m.  59.418. 
lOh.  7  m.  40.24  s. 

40.26  s. 
40.55  s. 
40.83  s. 

0.29  s. 
0.28  s. 

0.285s. 

»'      1 
In  this  case  -,  =  x  ,    &  =  lOh.  6m.  18.86s.,  Di  =  40.55s.,  and  Dj  =  0.285s. 
q       Z 

The  answer  is  lOh.  6m.  39.1s. 

5.  To-day,  July  1st,  at  noon,  the  moon's  declination  (distance 
from  the  celestial  equator)  is  6°  38'  10".8  north ;  at  4  o'clock  it  will 
be  5°  45'  51".3  ;  at  8  o'clock,  4°  53'  7".8 ;  at  midnight,  4°  0'  2".8 ;  and 
at  4  o'clock  in  the  morning  it  will  be  3°  6'  38".7.  Interpolate  for 
the  intermediate  hours. 


♦  In  tliis  oxamplethe  argument  is  the  time,  and  the  function  is  the  Right  Ascension,  i.  «., 
the  Right  Ascension  id  a  fuuction  of  the  time. 


292  APPENDIX. 


SECTION  11. 

PERMUTATIONS. 

328.  Combinations  ure  the  different  groups  which  can  be 
made  of  m  things  taken  n  in  a  group,  n  being  less  than  m. 

III. — Taking  the  5  letters  a,  h,  c,  d,  e,  we  have  the  10  following  cojnbi nations 
when  the  letters  are  taken  3  in  a  group,  or,  as  it  is  usually  expressed,  taKen  3 
and  3 :  abc,  abd,  ahe,  acd,  ace,  ode,  bed,  bee,  bde,  ede.  Taken  2  and  2,  we  have  the 
following  10  combinations  :  ab,  ac,  ad,  ae,  be,  bd,  be,  cd,  ee,  de.  It  is  to  be  notieed 
that  no  tico  combinations  eontain  tlie  same  letters;  t.  e.,  they  are  different  groups. 

329,  J^ermutations  are  the  different  orders  in  which  things 
can  succeed  each  other. 

III. — Thus  the  two  letters  a,  b  have  the  two  permutations  ab,  ba.  The  three 
letters  a,  b,  c  have  the  6  permutations  abc,  acb,  cab,  bac,  bcu,  cba. 

330.  Arrauffenients  are  permutations  of  combinations. 

III. — Taking  the  10  combinations  of  5  letters  taken  3  and  3,  and  permuting 
each  combination,  we  get  the  arrangements  of  5  letters  taken  3  and  3.  Thus 
the  combination  abc  gives  the  C  arrangements  abc,  acb,  cab,  bac,  bca,  cba.  In  like 
manner  each  of  the  10  combinations  of  5  letters  taken  3  and  3  will  give  6  arrange- 
ments ;  whence,  in  all,  5  letters  taken  3  and  3  have  60  arrangements. 

331,  I^roiJ. —  The  number  of  Arraiigements  of  m  things  taken 
n  and  n  is 

m  (m  -  1)  (m  -■  2)  (m  -  3) (m  -  n  +1). 

Dem. — Let  us  consider  the  number  of  arrangements  which  can  be  made  of  the 
m  letters  a,  b,  c,  d,  etc.,  taken  2  and  2.  Letting  a  stand  first,  we  can  have  ah,  ac, 
ad,  etc.,  to  w  —  1  arrangements.  Letting  b  stand  first,  we  can  have  ba,  be,  bd, 
etc.,  to  w  —  1  arrangements.  Thus  taking  each  of  tlie  m  letters  in  turn  we  can 
have  m  —  1  arrangements  in  each  case,  or  m  (m  —  1)  arrangements  in  all. 

Again,  eac7i  of  these  m(m  —  1)2  and  2  arrangements  will  give  m  —  2  arrange- 
ments  3  and  3,  by  placmg  before  it  each  of  the  letters  not  involved  in  it.  Thus 
we  have  m{m  —  l){m  —  2)  arrangements  of  m  letters  taken  3  and  3. 

Once  more,  ea^ih  of  these  m{m  —  1)  {m  —  2)  3  and  3  arrangements  will  give 
m  —  3  arrangements  4  and  4,  by  placing  before  it  each  of  the  letters  not  involved 
in  it.  Thus  we  have  m{m  —  1)  {m  —  2)  {m  —  3)  arrangements  of  m  letters  taken 
4  and  4. 

Finally,  we  observe  the  law  ;  i.  e.,  the  number  of  arrangements  is  equal  to 


PERMUTATIONS.  293 

the  continued  product  of  m{m  —  1)  {m  —  2)  {m  —  S)  -  -  -  -   {m  —  (n  —  l)}  or 
m(m  —  1) (m  —  2) (m  —  B)  -  -  -  -  {jn^n  +  1). 

332,  Cor.  1. — The  number  of  Permutations  o/m.  things  is 

1.2.3.4 m. 

This  is  evident  since  arrangements  become  permutations  when  the  number  Tn 
a  group  is  equal  to  the  whole  number  considered  ;  i.  e.,  when  n  =  m. 

333,  Cor.  2. — If'  p  of  the  m  letters  are  alike  {as  each  a),  q  otKefs 
alike,  r  others  alike,  etc.,  the  number  of  permutations  is 

1.2.3-4 m 

|p  X  [q  X  [r  X  etc.  * 

Thus  consider  the  permutations  of  a,  b,  c,  d,  viz.,  abed,  bacd,  acdb,  bcda,  acbd, 
bead,  aMc,  bade,  adcb,  bdca,  etc.  Suppose  b  to  become  a,  then  since  for  any  par- 
ticular position  of  c  and  d,  as  in  abed,  there  are  as  many  permutations  of  the  foifr 
letters  as  there  can  be  permutations  of  the  two  letters  a  and  b,  viz.,  1  x  3  ;  if  & 
becomes  a  there  will  be  1  x  2  fewer  permutations  when  these  two  letters  are 

alike  than  when  they  are  different,  i.  e.,       ~'     —  . 

So,  in  general,  if  p  of  the  letters  are  alike,  there  will  be  1-3 -3  -  -  -  -  jp,  or 
[p  fewer  permutations  than  if  they  are  all  different,  etc. 

334,  Cor.  3. —  T7ie  number  of  Com,binations  of  m  things  taken 

n  and  n  is 

m  (m  —  1 )  (m  —  2)  (m  —  3) (m  —  n  +  1) 

_________  . 


Since  arrangements  are  permutations  of  combinations,  the  number  of  ar- 
rangements of  w  things  taken  n  and  n  is  equal  to  the  number  of  combinations 
of  ni  things  taken  n  and  n  multiplied  by  the  number  of  permutations  of  n 
things.  Hence  the  number  of  combinations  is  equal  to  the  number  of  arrange- 
ments of  m  things  taken  n  and  n  divided  by  the  number  of  permutations  of  n 
things. 

Examples. 

1.  How  many  permutations  can  be  made  of  the  letters  in  the  word 
marble?     Of  those  in  A o m e .^     0^  iho^Qmlog arithms? 

2.  How  many  arrangements  can  be  made  of  10  colors  taken  3  and 
3  ?  Of  7  colors  taken  2  and  2  ?  Taken  3  and  3  ?  4  and  4  ?  5  and 
5  ?  6  and  6  ?  7  and  7  ?  How  many  mixtures  in  each  case,  irre- 
spective of  proportions  ? 

3.  How  many  different  products  can  be  made  from  the  9  digits 
taken  2  and  2  ?     3  and  3  ?     4  and  4  ?     5  and  5  ?     6  and  6  ?     7  and 

?     8  and  8  ?    9  and  9  ? 


294  APPENDIX. 

4.  How  many  different  numbers  can  be  represented  by  the  9  digits 
taken  2  and  2  ?     3  and  3  ?     4  and  4  ?  etc. 

5.  In  a  certain  district  3  representatives  are  to  be  elected,  and  tliere 
are  6  candidates.  In  how  many  different  ways  may  a  ticket  be  made 
up? 

6.  There  are  7  chemical  elements  which  will  unite  with  each  other. 
How  many  ternary  compounds  can  be  made  from  them  ?  How  many 
binary  ? 

7.  How  many  different  sums  of  money  can  be  paid  with  1  cent,  1 
3-cent  piece,  1  5-cent  piece,  1  dime,  1  15-cent  piece,  1  25-cent  piece, 
and  1  50-cent  piece  ? 

SuQ. — If  taken  1  and  1,  how  many  ?  If  2  and  2,  how  many  ?  If  3  and  3,  etc.? 
How  many  in  all  ? 

8.  In  how  many  ways  can  13  ladies  and  12  gentlemen  arrange 
themselves  in  couples  ? 

9.  If  you  are  to  select  7  articles  out  of  12,  how  many  different 
choices  have  you  ? 

10.  How  many  different  sums  can  be  made  from  1,  2,  3,  4,  5,  6, 
taken  2  and  2  ? 

11.  How  many  permutations  can  be  made  from  the  letters  in  the 
word  possessions?  (See  333,)  How  many  from  the  letters 
in  the  word  consistencies? 

12.  How  many  different  signals  can  be  made  with  10  different- 
colored  flags,  by  disi:)laying  them  1  at  a  time,  2  at  a  time,  3  at  a  time, 
etc.,  the  relative  positions  of  the  flags  with  reference  to  each  other 
not  being  taken  into  account  ? 


Probabilities. 

335.  TJie  Mathematical  I^rohdbility  of  an  event  is  the 
number  of  favorable  opportunities  divided  by  the  whole  number  of 
opportunities.  TJie  Mathematical  Iniprohahility  is  the  number  of  un- 
favorable opportunities  divided  by  the  whole  number  of  opportunities. 

III. — A  man  draws  a  ball  from  a  bag  containing  5  white  and  2  black  balls  ; 
the  opportunities  favorable  to  drawing  a  white  ball  are  five,  and  the  whole  num- 
ber of  opportunities  is  seven  ;  hence  the  mathematical  probability  of  drawing 
a  white  ball  is  \.    The  mathematical  improbability  of  drawing  a  white  ball  is  \. 


PROBABILITIES.  295 


Examples. 

1.  I  learn  that  from  a  vessel  on  which  my  friend  had  taken  pass- 
age, one  person  has  been  lost  overboard.  There  were  40  passengers, 
and  20  in  the  crew.  What  is  the  probability  that  my  friend  is  safe  ? 
"What  the  improbability  ?  If  I  learn  that  a  passenger  is  lost,  what 
then  is  the  probability  that  my  friend  is  safe?  What  that  he  is 
lost? 

2.  A  man  fires  into  a  flock  of  birds  of  which  6  are  white,  4  black, 
5  slate-colored,  and  3  piebald.  If  he  kills  one,  what  is  the  probability 
of  its  being  a  black  bird  ?  What  the  improbability  of  its  being  pie- 
bald ?  How  much  more  probable  is  it  that  he  will  kill  a  white  than 
a  piebald  bird  ?     A  black  than  a  piebald  ? 

3.  Twenty- three  persons  sit  around  a  table.  What  is  the  proba- 
bility of  any  given  couple  sitting  together  ? 

III.— Call  the  two  persons  A  and  B.  Then  wherever  A  may  sit,  there  are  22 
others  who  may  sit  beside  him  in  one  of  two  places  (on  his  right  or  left).  There 
are  therefore  2  favorable  and  20  unfavorable  opportunities. 

4.  What  are  the  odds  against  the  fourth  of  July  coming  on  Sun- 
day in  any  year  taken  at  random  ? 

SUG, — The  odds  against  an  event  is  the  ratio  of  the  unfavorable  to  the  favor- 
able opportunities. 

5.  The  moon  changes  about  once  in  7  days.  What  is  the  proba- 
bility that  a  change  of  weather  will  come  within  3  days  of  a  change 
in  the  moon  ? 

6.  The  letters  a,  e,  m,  n,  can  be  arranged  so  as  to  form  four  words, 
viz.,  mane,  mean,  name,  amen.  If  they  are  arranged  at  random, 
what  is  the  probabihty  of  their  forming  a  word?  What  the  "odds 
against "  their  forming  a  word  ? 

7.  Show  that  the  probability  that  a  leap-year  will  contain  53  Sun- 
days is  f . 

8.  Three  balls  are  to  be  drawn  from  an  urn  which  contains  5  ])lack, 
3  red,  and  2  white  balls.  What  is  the  probability  of  drawing  2 
black  balls  and  1  red  ? 

SuG's.— The  first  question  is,  How  many  opportunities  in  all  ?  That  is,  how 
many  different  groups  {combinations)  can  be  made  of  10  balls  taken  3  and  3. 
Second,  How  many  opportunities  favorable  to  drawing  two  black  balls  and  one 


296  APPENDIX. 

red  at  the  same  time  ?    There  are  5  black  balls,  and  these  can  be  combined  2  and 

5  4 

2  in  r-s  ,  or  10,  ways  ;  and  as  one  of  the  three  red  balls  can  be  obtained  in  3 
1  •  /* 

ways,  each  one  of  these  combined  with  one  of  the  10  ways  of  obtaining  the 

black  balls  will  give  10  x  3,  or  30,  favorable  opportunities  for  selecting  the  balls 

as  desired.     The  probability  is  therefore  -,^^o>  or  *• 

9.  If  from  a  lottery  of  30  tickets,  marked  1,  2,  3,  etc.,  4  tickets  are 
drawn,  what  is  the  probability  that  3  and  5  are  among  them  ?  What 
are  the  odds  against  it  ? 

Sug's. — From  30  how  many  combinations  of  4  and  4?  From  28  how  many 
combinations  of  2  and  2  ?    Odds  against  drawing  3  and  5,  143  to  2. 

10.  A  bag  contains  a  $5  bill,  $10  bill,  and  6  blanks.  What  is  the 
expectation  of  one  drawing  ?     That  is,  what  is  one  drawing  worth  ? 

SUG.— The  probability  that  one  draught  will  take  the  $5  bill  is  \,  and  hence 
is  worth  $^.  The  probability  that  the  $10  note  will  be  drawn  is  also  i,  and 
hence  this  expectation  is  %^^.  The  entire  expectation  is  therefore  $\-,  or 
$1.87^.  Hence  a  gambler  who  should  sell  such  chances  at  $2  each,  would  in 
the  long  run  make  money. 

11.  What  is  the  expectation  of  a  draught  from  a  bag  containing  5 
$2  bills,  4  $5  bills,  'Z  $10  bills,  1  $100  bill,  and  50  blanks? 

12.  In  a  given  bag  are  5  $2  bills,  3  $5  bills,  and  6  blanks.  What 
is  the  expectation  of  2  draughts  ? 

Suo's. — There  are  -     '    ■  =  91  opportunities,  or  ways  in  which  2  things  can 
1  •  »j 
be  drawn  from  14. 

5-4  .      ' 

There  are  :j — -  ways  in  which  $2  bills  rnay  be  drawn.    Hence  the  probability 

of  drawing  2  $2  bills  is  ^",  and  this  expectancy  is  $if. 

In  like  manner  the  probability  of  drawing  2  $5  bills  is  /, ,  and  this  expect, 
ancy  is  $3?. 

The  probability  of  drawing  2  blanks  is  ^f ,  and  this  expectancy  0. 

The  probability  of  drawing  1  $2  and  1  $5  bill  is  ^\,  and  this  expectancy  is 

$W- 

The  probability  of  drawing  1  $2  bill  and  1  blank  is  ^,  and  this  expectancy 
i8$l?- 

The  probability  of  drawing  1  $5  bill  and  1  blank  is  ^ ?,  and  this  expectancy 
is  $1^. 

The  entire  expectancy,  or  worth,  of  2  draughts  is  therefore  ^1  +  l^+^^-i-  +  f?- 
-f  1^  dollars,  or  $3.57f . 

Observe  that  the  sum  of  all  the  probabilities,  i.  6.,  \^  +  g^  -H  i  }  +  i.f  +  y^  +  at, 
is  1,  as  it  should  be. 


PROBABILITIES.  297 

That  the  probabilit}^  of  drawing  1  $3  bill  and  1  $5  is  ^f,  is  seen  thus  :  There 
are  5  opportunities  favorable  to  drawing  1  $2  bill,  and  with  each  of  these  there 
arc  3  opportunities  favorable  to  drawing  1  $5  bill. 

13.  There  are  4  white  balls  and  3  black  ones  in  one  bag,  and  2 
white  ones  and  7  black  ones  in  another  bag.  What  is  the  probability 
of  drawing  a  white  ball  from  each  bag  at  the  first  draught  from 
each  ? 

Solution.— There  arc  in  all  7  opportunities  of  drawing  a  ball  from  the  first 
bag,  and  with  each  one  of  these  there  are  9  opportunities  from  the  second 
bag  ;  hence  there  are  7  x  9,  or  C3  opportunities  in  all.  Again,  there  are  4  favor- 
able opportunities  for  drawing  a  white  ball  from  the  first  bag,  and  with  each  of 
these  there  are  2  favorable  opiwrtunities  for  drawing  a  white  ball  from  the 
Eccond  bag;  t.  e.,  there  are  in  all  4  x  2,  or  8,  favorable  opportunities.  Hence 
the  probability  is  -^.i.  Notice  that  this  compound  probability  is  the  product  of  the 
two  simple  probabilities. 

14.  The  probability  that  A  can  solve  a  problem  is  f,  and  that  B 
can  do  the  same  is  \,  what  is  the  joint  probability  ? 

Sug's. — The  student  will  observe  that  there  are  4  possible  results,  viz.  : 
1.  Both  may  succeed,  of  which  the  probability  is  -^- ;  2.  A  may  succeed  and  B  fail, 
of  which  the  probability  is  f,!  ;  3.  5  may  succeed  and  A  fail,  of  which  the  prob- 
ability is  -/a-;  and  4.  Both  may  fail,  of  which  the  probability  is  \%.  Now  either 
the  first,  second,  or  the  third  result  will  give  a  solution.  Hence  the  probability 
of  success  is  ^  +  H  +  a*s  =  §f .  or  \. 

This  result  may  be  more  expeditiously  obtained  by  considering  that  they 
will  succeed  if  both  do  not  fail.  The  probability  of  J.'«  failure  is  |,  and  of  B's  y. 
Hence  the  probability  that  both  will  fail  is  ^  x  f ,  or  ?^;  and  the  probability  of 
success  is  1— f,  or  y. 

15.  It  may  be  said  that  on  an  average  10  persons  will  die  during 
the  next  10  years 

Out  of  every  G2  whose  present  age  is  30, 

«  a        45  a  a       40, 

««        «      35  "  ''      50, 

«         «       25  "  "      60. 

What  is  the  probability  Unit  a  person  who  is  30  will  live  till  he  is 
60  ?     What  that  a  person  who  is  40  will  live  till  he  is  70  ? 

StjG's.— Let  ua  examine  the  probability  that  the  man  who  is  30  will  die  before 
he  if  OJ.  The  probability  that  ho  dies  before  40  is  |i^,  and  that  he  lives  to  40 
f  |.  Now  the  probability  that  a  man  who  is  40  dies  before  50  is  ii  Hence  the; 
probability  i.i  -^^  of  §i  that  this  man  lives  to  40  and  dies  between  40  and  50,  or 
it  Is  5i  of  ^^1  that  ho  lives  to  50.     Finally,  ^le  probability  that  he  dies  between 


298  ATPLNDIX. 

50  and  60  is  ^  of  ^  of  H,  or  »t  is  U  of  H  of  B  ^liat  he  lives  from  50  to  6a 
Hence  the  probability  that  a  man  who  is  30  will  die  before  he  is  60  is 

i^  +  B  X  -L^  +  B  X  ^^  X  H,ori^g; 

and,  consequently,  the  probability  that  he  will  live  is  1  —  ^^,  or  i^§;  i.  e.,  it  is 
a  little  more  probable  tliat  a  man  who  is  30  will  die  before  he  is  60,  than  that 
he  will  live  to  60. 

16.  What  is  the  probability  that  two  persons,  A  and  B,  aged  re- 
spectively 30  and  40,  will  be  alive  10  years  lience  ? 

SUG'S.— Chance  of  A's  being  alive  ^,  of  B'a  U,  of  both  ^  x  J^,  or  ^^. 


LOGARITHMS  OF  NUMBERS. 


N. 

Log.    1 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

O'OOOOOO 

26 

I. 414973 

51 

1.707570 

7(] 

1-880814 

2 

0'3oio3o 

27 

i.43i364 

52 

1.716003 

U 

1-886491 

8 

0.477121 

23 

1.447158 

53 

1-724276 

73 

1-892095 

4 

o.6o2o6o 

2'J 

1.462393 

54 

1.732394 

79 

1.897627 

5 

0-698970 

80 

i-477'2i 

55 

1-740363 

SO 

1-903090 

6 

0.778151 
0.845098 

81 

1.491362 

56 

I -748188 

81 

1-908485 

7 

82 

i.5o5i5o 

57 

1.755875 

82 

i-9i38i4 

8 

0.903090 

83 

I.5i85i4 

58 

1-763428 

83 

1-919078 

9 

0.954243 

84 

1.531479 

59 

1.770852 

84 

1-924279 

10 

I'OOOOOO 

85 

1.544068 

60 

i-778i5i 

85 

I -929419 

11 

i.o4i3o3 
1-079181 
I- 1 13943 

86 

I -556303 

61 

1-785330 

86 

1-934498 

12 

37 

1.568202 

62 

1-792392 

87 

1-939519 

13 

88 

1-579784 

63 

I -799341 

88 

I-9444C3 

14 

1.146128 

89 

1-591065 

64 

1-806180 

89 

1-949390 

15 

1-176091 

40 

1-602060 

1  g:> 

1-812913 

90 

1-954243 

IG 

1-204120 

41 

1-612784 

I  66 

1-819544 

91 

1-9590:1 

17 

1-23044Q 

1-255273 
1-278754 

42 

1-623249 

1  67 

1-826073 

92 

1.963783 

18 

43 

1 -633468 

i  68 

1-832509 

!3 

1 .968483 

19 

44 

1-643453 

i  69 

1-838849 
1-845098 

94 

1.973128 

20 

I -301030 

45 

1-653213 

70^ 

95 

1.977724 

21 

1-322219 

46 

1-662758 

71 

1-851258 

96 

I -982271 

22 

1-342423 

47 

1-672098- 

7-J 

1-857333 

97 

1.986772 

23 

1.361728 

43 

1-681241 

73 

1.863323 

9^ 

I -991226 

24 

1.3802 1 1 

49 

1-690196 

74 

1-869232 

99 

1-995635 

25 

1.397940 

50 

1.698970 

75 

1 

1-875061 

100 

2-000000 

Remark. — In  the  following  Table,  the  frst  two  fgurea^  in  the  first  column  of 
Logarithms,  are  to  be  prefixed  to  each  of  the  numbers,  in  the  same  horizontal 
line,  in  the  next  nine  columns;  but  when  a  point  (•)  occurs,  a  0  is  to  bo  put 
in  its  place,  and  the  two  initial  fjurcs  are  to  be  taken  from  the  next  line  below. 


SCO 


LOGARirnMS  cir  numbers. 


K. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

I). 

100 

101 
102 

103 
104 
105 
106 
107 
108 
109 

4321 
86oo 

012837 
7o33 

021180 
53o6 
o384 

033424 
7426 

0434 
4751 
9026 
3259 
745 1 
i6o3 
57.5 

nit 
7S25 

0868 
5i8i 
9451 
368o 
7868 
2016 
6i25 
•195 
4227 
8223 

i3oi 

98^ 
4100 
8284 
2428 
6533 
•600 
4628 
8620 

1734 
6o38 
•3oo 
452. 
8700 
2841 
6942 
1004 
5029 
9017 

2166 
6466 

•724 
4940 
91 16 

3252 

7350 
1408 

5430 
9414 

2598 
6894 
1147 
536o 
95J2 
3664 

1812 
583o 
981 1 

3029 
7321 
1570 
5779 
9947 
4075 
8164 
2216 
623o 
•207 

3461 

7748 
1993 
6197 
•36 1 
4486 
8571 
2619 
6629 
•602 

3891 
8174 
24i5 
6616 

8978 

3o2i 

7028 

•998 

4.12 
428 
424 
419 
4.6 
412 
4o3 
404 
400 
396 

110 
111 
112 
lid 
114 
115 
116 
117 

lis 

119 

041393 
5323 
9218 

053078 
6905 

060698 
44^8 
8186 

071882 
5547 

1787 
5714 
0606 

3463 

7286 

1075 
4832 
8557 
225o 
59.2 

2182 
oio5 

7666 

1452 
5206 
8928 
2617 
6276 

2576 
6495 
•380 
423o 
8046 
1829 
558o 
9298 
2983 
6640 

688? 
•766 
46i3 
8426 
2206 
5953 

7004 

3362 

ml 

2582 

6326 
••38 
3718 
7368 

3755 
7664 
1538 
5378 
9185 
2g58 
6699 
•407 
4o8d 
7731 

414S 
8o53 
1924 
5760 
9563 
3333 
7071 

2t 

8094 

4540 
8442 
2309 
6142 
0942 
3709 
7443 
1145 
4816 
8457 

4932 

8b3o 
2C94 
6524 

•320 

4o83 
7.S.5 
i5.4 
5.82 
8819 

393 
389 
386 

382 

l]l 
32 
369 
366 
363 

120 
121 
122 
123 
124 
125 
126 
127 
128 
129 

130 
131 
132 
138 
134 
135 
106 
137 
133 
139 

140 
141 
142 
143 

1:4 
U5 
1-16 
147 
143 
U9 

079 1 81 

082785 
636o 
0905 

093422 
6910 

100J71 
38o4 
7210 

110590 

o543 
3i44 
6716 
•258 
3772 
7257 
0715 
4146 
7549 
0926 

7071 
•611 
4122 
7604 
1059 
4487 
7^88 
1263 

•266 
3861 
7426 
•963 
4471 
79^1 
i4o3 
4828 

b2  27 

1599 

•626 
4219 

4B20 
82  >8 

1747 
5169 

8563 

1934 

XI 

8i36 
1667 
5169 
8644 
2091 
55io 
8903 
2270 

1347 
4934 
8490 
2018 
65i8 
8990 
2434 
585, 
9241 
26o5 

1707 
5291 
8845 
2370 
5866 
9335 

2777 
6191 
9579 
2940 

2067 
5647 
9198 
2721 
62i5 
0681 
3119 
653i 
C916 
3275 

2426 
6004 
9552 

3071 
6:62 
••26 
3462 
6871 
•253 
3609 

36o 
357 
355 
35i 

34? 
343 
340 
338 
335 

113943 

7271 
120574 
3852 
nio5 
i3o334 
3339 
6721 

i43oi5 

7603 
0903 
4178 

0655 
3858 
7037 
•194 
3327 

4611 

7934 

I23l 

45o4 
7753 
0977 
4177 
-354 
•5o8 
3639 

4944 

i265 

1:60 

41 3o 
8076 
.298 
4496 
7671 
•822 
3951 

5278 

8:95 

18.^8 

5i56 
fc3v9 
1619 
4814 
7987 
ii36 
4263 

56ii 
8926 
2216 
5481 
8722 

83o3 
1450 

4574 

5,43 
9256 

2)44 
5bo6 
9045 
2260 
5451 
8618 
1763 
4885 

6276 
9586 
287I 
6i3i 
9368 
258o 
5769 
8934 
2076 
5196 

6608 
9915 

6456 
9690 
2900 
6086^ 
9249 
2389 
5507 

6940 
•245 
3525 
6781 
••.2 
3219 
64o3 
9564 
2702 
58i8 

333 
33o 
328 
325 
323 

321 

3i8 
3i5 
3i4 
3u 

146128 

152288 
5336 

n^a 

i6i3o8 
4353 
7317 

170262 
3ifc6 

6438 
9527 
2594 
6640 
8664 
1667 
465o 
7^1 3 
0^35 
3473 

6748 
9S35 
2900 
6943 
8965 
1967 

4947 
7908 

0^48 
3769 

7o58 
•142 
32o5 
6246 
9266 
2266 
5244 
8203 
1141 
4060 

7367 
•449 
35io 
6049 
9:67 
2564 
5541 
8497 
1434 
435i 

76:6 
•7^^6 
38i5 
6S52 
9^-68 
i863 
583S 
8792 
1726 
4641 

79S5 
io63 
4120 
7'54 
•i63 
3i6i 
6.34 
9086 
2019 
4932 

IJ70 
4424 

7457 
0469 
3460 

643o 
9380 

23  II 
5222 

86o3 

1676 
4728 

7759 
•769 
3758 
6726 
9674 
26o3 
55i2 

891 1 
1982 
5o32 
8061 
1068 
4o55 
7022 
9968 
2895 
58o2 

309 
307 
3o5 
3o3 
3oi 
299 
297 
295 
293 
291 

150 
151 
152 
153 
154 
1.-5 
156 
157 
158 
159 

176091 
8977 

181844 
4691 
7521 

190332 
3.25 

5899 

8657 

201397 

633i 
9264 
2129 
4975 
7fco3 
0612 
34o3 
6176 
8932 
1670 

6670 
9552 
240 
5259 
8084 
0802 
368 1 
64^3 
9206 
1943 

6959 
9S39 
2700 
5542 
8366 

39^9 
6729 
9481 
2216 

7248 
•126 
2985 
5825 
8647 
I45l 
4237 
7co5 
9755 
2488 

7536 
•4i3 
3270 
6108 
8928 
1730 
4314 
7281 
•C29 
2761 

7825 
«-699 
3555 
tlgl 
9209 
2010 

•3o3 
3o33 

8m3 
•985 
3839 
6674 
9490 
2289 
5069 
7832 
•577 
33o5 

8401 
1272 
4.23 

6936 

977  • 
2D67 
5346 
8107 
•85o 
3577 

8689 
1558 
4407 
7239 
••5 1 
2846 
5623 
8382 
1 1 24 
3846 

2P9 

287 
285 
283 
281 
279 

276 
274 

272 

N. 

0 

1  1  2 

8 

4 

5 

6    7 

8 

9 

D. 

LOGATJTmrS   OF  KTTMBERS. 


301 


N. 

0 

1 

?. 

3 

4 

5 

6 

y 

8 

9 

D. 

160 

304130 

4391 

4663 

4934 

5204 

5475 

5746  6016 

6286 

6556 

271 

161 

6826 

7096  7J65  i  76J4 

7904 

8173 

8441   8710 
1121   i383 

8979 

9247 

269 

162 

95i5 

978J  ••5i  I  •J  19 

•586 

•853 

;654 

1921 

267 

163 

312188 

2454 !  2720    29% 

3252 

35iS  3783  1  4049  i  43i4  1 

4379 

266 

164 

4844   5109  ;  5373  1 

5b  J  3 

5902 

6166 

6430  6694 

6957 

7221 

264 

165 

7484 

7747 

8010 

8273 

8d36 

8793 

9060  9323 

9585 

9846 

262 

166 

320108 

0370 

o63i 

0892 

n53 

1414 

1675  1936 

2196 

2456 

261 

107 

2716 

5309 

^9"^^ 

3236 

3496 

3755 

4oi5 

4274 

4533 

4792 
7372 

5o5i 

259 

168 

5D68 

5826 

6084  i 

6342 

6600 

6858 

7.15 

763o 

258 

169 

7887 

8144 

8400 

8657 

8913 

9170 

9426 

9682 

9938 

•193 

256 

170 

230449 

0704 

0960 

I2l5 

1470 

1724 

:?;? 

2234 

2488 

2742 

254 

171 

325o 

35o4 

3757 

4011 

4264 

4770 

5o23 

5276 

253 

173 

5d28 

5781 

6o33 

6285 

6537 

6789 

7041 

7292 

7544 

7795 

252 

173 

8046 

8297 

8548 

8799 

T5tl 

9299 

9550 

9800 

••5o 

•3oo 

25o 

17t 

340349 

0799 

1048 

1297 

1795 

2044 

2293 

2541 

2790 

1% 

175 

3o38 

3286 

3534 

3782 

4o3o 

4277 

4525 

4772 

5019 

5266 

176 

55i3 

5759 

6006 

6252 

6499 
8934 
1395 

6745 

6991 

7237 

7482 

772S 

246 

177 

173 

7973 

25o420 

8219 
0664 

8464 
0903 

8709 

Ii5i 

Vefs 

9443 

1881 

9687 

2125 

l^ll 

•176 
2610 

243 

243 

179 

2853 

3096 

3333 

358o 

3822 

4064 

43o6 

4548 

4790 

5o3i 

242 

180 

255273 

55i4 

5755 

5996 

6237 

6477 

6718 

6958 
9355 

7198 

7439 

241 

ISl 

7679 

70.8 
o3io 

8i58 

83o3 

8637 

8877 

1263 

91 16 

9594 

9833 

239 

183 

260071 

0343 

0787 

1025 

i5oi 

1739 

1976 
4346 

2214 

233 

183 

245i 

26S8 

2925 

3i62 

3399 

3636 

3873 

4109 

4582 

237 

1S4 

43i8 

5o54 

6290 

5525 

5761 

^t 

6232 

6467 

6702 

6937 

235 

185 

7172 

7406 

7641 

7875 

8110 

8578 

8812 

9046 

9279 

234 

1S6 

95i3 

9746 

9980 

•2l3 

•446 

•679 

•912 

1144 

1377 

1609 

233 

187 

271842 

2074 

23o6 

2533 

2-'70 

3ooi 

3233 

3464 

360 

3927 

232 

183 

4.58 

4389 

4620 

485o 

5o8i 

53n 

5542 

5772 

6002 

6232 

23o 

189 

6462 

6692 

6921 

7i5i 

7330 

7609 

7838 

8067 

820 

8525 

229 

H'O 

278754 

8982 

0211 

9439 

9667 

9895 

•123 

•35i 

0578 

•806 

223 

191 

28io33 

1261 

14S8 

1715 

1942 

2169 

2396 

2622 

2849 

3075 

227 
226 

193 

33oi 

3327 

3753 

3979 

42o5 

443i 

46D6 

4882 

5 1 07 

5332 

103 

5557 

5782 

6007 

6232 

6456 

663 1 

6905 

7i3o 

7354 

7573 

225 

194 

7802 

8026 

8249 

8473 

8696 

8920 

9143 

9366 

f,l 

9812 

223 

105 

2QOo35 

0257 

2478 

0480 

0702 

0925 

1147 
3363 

1369 

i59t 

2o34 

222 

106 

2256 

2699 

2920 

3i4i 

3584 

3jo4 

4025 

4246 

221 

107 

4466 

46S7 

4907 

6127 
7323 

5347 

5567 

5787 

8198 

6226 

6446 

220 

103 

6665 

68S4 

7104 

7542 

7761 

7979 

8416 

8633 

219 

199 

8853 

9071 

9289 

9507 

9725 

9943 

•161 

•378 

•595 

•81 3 

218 
217 

200 

3oio3o 

1247 

1464 

1681 

1898 

2114 

233i 

2547 
4706 

2764 

2980 

201 

It 

34. i 

3623 

3  84  4 

4059 

4275 

4491 

4921 

5i36 

216 

203 

5566 

5781 

5996 

6211 

6425 

6wJ9 

8778 

6854 

7068 

72S2 

2l5 

203 

7496 

7710 

792  i 

8137 

835i 

8364 

8991 

9204 

9417 

2l3 

204 

030 

9843 

••56 

•263 

04S. 

•693 

•906 

1118 

i33o 

1 542 

212 

205 

311754 

1966 

2177 

2389 

2600 

2812 

3o23 

3234 

34 ',5 

3656 

211 

206 

3J67 

4078 

4289 

4499 

47«o 

4920 

5i3o 

5340 

555i 

5760 
7854 

210 

207 

5970 

6180 

6390 

nv. 

6809 
8893 

7018 

7227 

7436- 

764^ 

209 

203 

8o63 

8272 

8481 

0106 

9314 

9522 

973  d 

9933 

203 

209 

320146 

o354 

o562 

0769 

0977 

Ti84 

1391 

1598 

i8o5 

2012 

207 

210 

322?I9 

24^6 

2633 

2839 

3046 

3252 

3458 

3665 

f^l 

j'^V 

306 

£11 

42';2 

44.3 

4694 

'^?o 

5io5 

53io 

53i6 

5721 

5926 

6i3i 

2o5 

£12 

6336 

6541 

6745 

7155 

If^ 

7563 

7767 

7972 

8176 

204 

213 

833o 

85:^3 

8787 

8991 

9194 

9601 

9805 

•••3 

•211 

203 

214 

33o4i4 

0617 

0819 

1022 

1225 

1427 

i63o 

i832 

2o34 

2235 

202 

215 

2438 

2640 

2842 

3o44 

3246 

3447 

3649 

385o 

4o5i 

4253 

202 

216 

4434 

4655 

4856 

5o57 

52  57 

5453 

5658 

58  "9 

6059 
8o53 

6260 

201 

217 

6460 

6660 

6860 

7060 

7260 

7459 

7659 

73:3 

8257 

200 

213 

8456 

8656 

8855 

9054 

9253 

945i 

96:50 

9849 

••47 

•24'^ 

199 

219 

340444 

0642 

0841 

io39 

i[237 

1435 

1632 

i83o 

2023 

2225 

193 

N. 

i  ^ 

1 

2 

3 

4 

5 

G 

7 

8 

9 

H 

3C2 


LOGARITHMS   OF  NUMBERS. 


N. 

0 

1 

2 

3 

4 

6 

6 

7 

8 

9 

I). 

220 
221 
222 
228 
224 
225 
226 
227 
£28 
229 

342423 

63?3 
83o^ 
35o24S 
2i83 
4ioS 
6o25 

'^1 

2620 
4589 
5549 
85oo 
0442 
2375 
43oi 
6217 

8l2D 
••25 

2817 
47M 
6744 
869i 
06J6 
2568 
4493 
6408 
83i6 

•2l5 

3oi4 
4981 

^l 

0829 

2761 
4685 

Its 

•404 

3212 

5178 

7t35 
9083 

1023 

2q54 
4S76 
6790 
8696 
•593 

3409 
5374 
7330 
9278 
1216 
3i47 
5o68 
6981 
8886 
•783 

36o6 
5570 
7525 
9472 
1410 
3339 
5260 
7172 
9076 
•972 

3802 
5766 
7720 
9666 
i6o3 
3532 

5432 

7363 
9266 
1161 

3999 
59^2 

9860 
1796 
3724 
5643 
V5i 
94j6 
i35o 

8110 
••54 
,980 
3916 

5834 
774  i 
9646 
I539 

;^ 

193 
194 
193 
193 
192 
19. 

:?; 

230  1 

231  1 

232  ! 
253 
2.i4  i 
235 
23ti 
237 
238 
239 

361728 
36i2 

5488 
7356 

Q2l6 

37io68 
2912 

4748 

5675 
7542 
0401 
1253 
3oq6 
4932 

2io5 

3988 
5862 
7729 
9587 
1437 
3280 
5ii5 
6942 
8761 

2294 

4176 
6040 
79i5 
9772 
1622 
3464 
5298 

8943 

2482 
4363 
6236 
8roi 
9958 
1806 
3647 
5481 
7306 
9124 

2671 
455 1 
6423 

8287 
•143 

3?3i 
5664 
7488 
9306 

2859 
4739 
6610 
8473 
•323 
2.75 
401 5 
5846 
7670 
9487 

3048 
4926 
6796 

236o 
4198 
6029 
7832 
9668 

3236 
5. .3 

6983 
8845 
•698 

2344 

4382 
62.2 
8o34 
9849 

3424 
53oi 
7169 
9o3o 
•883 
2728 
4565 
6394 
8216 
••3o 

.837 
3636 
5428 
72.2 
8989 

:?^? 
4277 
6025 
7766 

188 

ii 

.S5 

.84 
184 
.83 
182 
181 

240  t 
211  i 
242 
243 

2u  ; 

24-,  ! 
246 
247  1 
24^  ; 
249 

380211 
2017 
3Si5 
56o6 

9166 
390935 

:% 

6199 

0392 

^ 

5785 
7568 
9343 
1112 
2873 
4627 
6374 

0573 
2377 

5^64 
7746 
9520 
1288 
3048 
4802 
6548 

III', 

4353 
6142 

9^>98 
1464 

3224 

4977 

6722 

0934 
2737 
4533 
6321 
8101 
9875 
1641 
3400 
5i52 
6896 

iii5 

2917 
4712 
6499 

"^? 

53a6 

7071 

1296 
3097 
4891 
6677 
8456 
•228 

It 
55o. 
7245 

1476 
3277 
D070 
6856 
8634 
•4o5 
2169 
3926 
5676 
7419 

1 656 
3436 
5249 
7034 
8^1. 
•582 

2343 

4101 
585o 
7592 

.81 

.80 
'79 

178 
.78 

177 
.76 
.76 
175 
174 

250  ' 

251 

252 

258 

2.54 

256 

256 

257  ! 

2--.a  ; 

25D 

397940 

9674 

401401 

3l2I 

4834 
6540 
8240 
9933 
411620 
3300 

8114 

is? 

6710 
8410 
•102 
1788 
3467 

8287 
••20 
1745 
3464 
5176 
6«8i 
8579 
•271 

8461 

•192 

Zl 

5346 
7o5i 

8749 
•440 
2124 
.'Wo3 

8634 
•365 
2089 
3807 
55.-! 
7221 
89.8 
•600 
2293 
3970 

8808 
•538 
2261 
3978 
5688 
7301 

2* 

2461 
4137 

8981 
•711 

2433 
4149 

5858 
756. 
9257 
•946 

9154 
•883 
26o5 
4320 
6029 
7731 
9426 
1114 
2796 
4472 

9328 
.056 

2777 
4492 
6.99 

5i 

2964 
4639 

95oi 
.228 
2949 
4663 
6370 
8070 
9764 
145. 
3.32 
4806 

.72 
171 
.71 

.70 
169 

!^ 
167 

260  i 
2»il 
262 
2«8 

264 ; 

265 
265 
267 
263 
269 

414973 
6641 
83oi 
9956 

421604 
3246 
4882 
65ii 
8i35 
9752 

5i4o 
6807 
8467 
•121 
1768 
3410 
5o45 
6674 
8297 
9914 

5307 
6973 
8633 
•286 

52o3 
6836 

2i?? 

5474 

9. 
Si 

•236 

5641 
73o6 

2261 

7.61 
6783 
•398 

58nR 
7472 

2426 
4o65 
5697 
7324 
8944 
•i59 

5974 
7638 
9295 

4228 

586o 
7486 
9106 
•720 

6141 

7804 
9460 
mo 
2754 
4392 
6023 
7648 
9268 
288. 

63o8 

lilt 
.275 
2918 
4555 
6186 
78.. 
9429 
1042 

6474 
8i35 

979' 
1439 
3o82 
4718 
6340 
7973 
9591 

I203 

.65 
.65 
.64 
.64 
i63 
162 
.62 
161 

270  , 

271 

£72 

278 

274 

275 

276 

277 

278 

'  279 

1 

43 1 364 
616? 

4045 

.56o4 

i525 

3i3o 
4729 

6322 

7909 
9491 
1066 
2637 
4201 
5760 

i685 
3290 
4888 
6481 
8067 
9648 
1224 
2793 
4357 
59.5 

1846 
345o 
5048 
6640 
8226 
9806 
T381 
2950 
4Di3 
6071 

2007 
36io 
5207 

nt 

3io6 
4669 
6226 

2167 
3770 
5367 
6957 
8542 
•122 
1695 
3263 
4825 
6382 

2328 

7II6 

8701 
•279 

i852 
3419 
49B1 
6537 

2488 
4090 
5685 

&59 
•437 
2009 
3576 
5.37 
6692 

2649 
4249 

5844 
7433 

2166 
3732 
5293 
6848 

2809 
4409 
6004 
7592 
9.75 

•732 
2323 
3889 
5449 
7003 

.6. 
160 

,59 
.59 
i58 
i58 

■H 

i55 

|. 

0 

1 

2 

8 

' 

5 

6 

7 

8 

9 

1).  ' 

ANSWERS. 


PARTI. 


[Note.— The  full-faced  figures  in  connection  with  the  nnniber  of  the  page  refer  to  theArti- 
eles  in  the  text.  The  numbers  in  parenthesis  in  the  paragraph  refer  to  the  particular  Example. 
*  *  *  indicate  that  it  is  not  thought  expedient  to  give  the  anewer.] 


ADDITION. 

(Page  13,  68.) 
(1.)  -la.         (2.)  Aa^-b--^a'^b^ab'-^'Zb'-^a\  (3.)  15ca^x^-^2ba^x^-h 

Irnx'^y^.        (4.)  Az^-x^-^-bx^+Tyx^y-db-x^-^.       (5.)  x.       (6.)  hcz~-x^-\-2z^. 
(7.)    {a^c)z-\-{m-ba)y  (8.)    (2rt+2&+8c-2^+^-2»>zj^+(13rt-4-4;i-f-2^yi 

(9.)  {a-^b+\)x*-¥(J>-a-^\)xy+{a-b+\)y\        (10.)  {a+m){x-^y)+{h-n){x'-y). 
(11.)  8(m+n-2)V^^.  (12.)  «.T;"^-H(3-m)^-i  +  3c.  (13.)    ^Va'-x'. 

(U.)  0.      (15.)  lx^+\y'^-h^--'.      (10.)  ia+b+c)  V^^^^.      (17.)  2(«-2m).r^ 
+3(m— l)y*+8c2. 


SUBTRACTION. 

(Page  15,  7^.) 

(2.)  ic3+a;«-2aj-a  (S.)  -2(a;«+flw).  (4.)  6.#+2;t5^.         (o.)  4»V^. 

(0.)  10  ^T+x^-QayK      (7.)  rt(^«-^)+(10-;r)  4^.      (8.)  bx{X'-h)-4:  ^^-f  2. 
(9.)    2  4/«j^-Va6.  (10).    a-^b+c-fd',    5(t;    a-'b+c-d-,    and    8rt+9A, 


(11.) 


«  #  # 


30 1  ANSWERS. 

M  ULTIBLICATION. 

(Page  20,  87,) 
(1.)  na^bx'^y^.  (2.)    24»i'+>r''^"^--\  (3.)    lOOx^^'^y^  ;  and  -  ^ah^ 

(4.)  w"  ;  1;  a«  ;  w"^^;  rt^^;  c''^*.  (5.)  3a*-+-10a6-86*.  (6.)  a;*+a;'y«+y^ 
(7.)  m^'  — rw'^t'o'  +  ^i^+o^.  (8.)  rt^— «"'+»+«"•+«- a^+'+r^^+'—tt*.  (9.)  «*  — 
('^+6+c+f02''  +  (nh+ac+lK-\-ad+hd-{-cd)z''  —  {abc  +  abd  +  acd  +  6crf)2  +  abed. 
(10.)  x^-ys.  {\\.)  a^b  i*+\.  (12.)  20«6«+30a''--^^»&'^+»+> 

— 10a^-*+r-»y-«-'--15a*+/'-t-'.  (18.)  *  *  ♦.  (14.)  *  *  * 

(15.)  -a*+2rt'6*-&<+2(a«+6«)c'-6*. 

(Page  21,  88.) 
(3.)  0«^-10a='a;-22^*«x«+46aiJ»-20t*.  (4.)  4^«-16a'»6*  +  10«^6'-f  15«6* 

-C5&*.        (5.)  a*-.T*.        (6.)  a;»-r,x*+10a;»--10j;«+5a;-l. 


DIVISION. 
(Page  24,  J05.) 
(1.)  mvV  /-^;  (^)^;  1,;   1,;  .J;  J,.  (2.)  ^;  -^^^  ;  ^!^. 

(3.)***.  (1.)   Last  two,   //-«"~;/-''-/,-/>i+'';a:^~'^'-2--i--%-3,t.-.. 

(5  to  II.)  *  •:>  ».  (12.)  (.r+y)«.  (18.)  *  •"  «.  (14.)  *  *  *  (15.)  a  +  b. 
(16.)  te'-- ••^r>+'f-rt««+2"-'^*o''+6''6"«.  (17.)  ?;»*+««•*.  (18.)  m^'4-??.a;*.-H 
»/i-  +  m.        (10.)  ;i,B«— 2.T  +  A-.        (20.)  a>^— «'y^+y^— ajV-yM-f  3^. 

(Page  20,  100.) 

(2.)  a;«-5^.r+4ri«.      (3.)  2rt''4-4rt«4-8«+16.      (4.)  %y*^AxK      (5.)  aj"-*"^ 

(Page  27,  i07.) 

(8.)  2y^~8y^'4-12v«-8y+2.  (4.)  :f.'^-^x'y'¥x*i/+:thf^x^y^+i'y''+y^\ 

1  +  ar  -f  x«  -f-  .r '  +  .e^  4-  »^  4-  x^  -I-  a;^  +0?^   ct-.  (5.)  *  ♦  *  ''-'  >  *  *  *. 

(7.)  •  ♦  ♦. 


ANSWERS.  305 


FACTOniNG. 

(Page  31,  122. 

Examples  in  factoring  are,  in  general,  of  such  a  nature  that  the  answers  can- 
not be  giveu  without  destroying  the  utility  of  the  problem  ;  hence  only  the  fol- 

l.»wing  are   given:  (23.)  k'y^—l^m^yh^+k'm^yK^-k^m^y^^-k^m^y^z^ 

-k'myH  +  mz".  (24.)  aj^— a;' V^^  +^' V— -^'V'^  +  ^ V— ^' V'^*  +y^ • 

(26.)***;     l-a=(l+  i/«)(l-|/«);    l+«isdivisiblebyl+  t'a,     1+ '^, 


GREATEST  OR  HIGHEST  COMMOX  DIVISOR, 

(Page  34,  124,) 

(1.)  12.         (2.)  12.         (3.)  3.         (5.)  2kH'mK         (6.)  2a'b.        (7.)  x^Y^z\ 
(S.)  Q?y.       (10.)  Ah^x-AbK 

(Page  38,  120.) 

(3.)  x-1.        (4.)  x+\.        (5.)  2a+3x.        (G.)  2«-6.         (7.)  4(ar»-2ary+y'). 
(8.)  2aa;=»-6ax*+10«a;-2«. 

(Page  38,  i50.) 

(1.)  2+6.        (2.)  2(a;+y). 


LOWEST  OR  LEAST  COMMOX  MULTIPLE. 

(Page  39,  132,) 

(2.)  («+&)Ma-&)^        (3.)  a-'^-4.        (4.)  4(a^-2a^+l).        (5.)  49r)9«-'&^'j:V^. 
(6.)  1-I8a4-81«^  (8.)  (a;^-39x+70)(.i:-10).  (0.)  (x-l)(.i-+2)(a:-3). 

(10.)  («='-4^^6+9aft*-10&')(a+4^).  (11.)  ir^-Ur'+71a:«-154a;+123. 

(12.)  2?''+7x*-10x'-70x*4-9.i-  +  03. 


306  ANSWERS. 

FRACTIONS. 
(Page  48,  167.) 
(1.)  ♦  *  *  (2.)  ♦  *  *.  (3.)  \+x+x^  +ir'4-  etc. ;     iC^+lO-  ~, ; 

m-\-n  2       4       8       16 

4- etc.;       a+x\       x»+l+ar-*+iC-'»H-a;-'"-h  etc. ;       1— wa-''»+w2a-4»— r^-^a-S" 
+  /i*a-8»-;t»a-i""+  etc.  (4.)  3  •  7-»a- V  ;  (m  +  ^iy+s .  c'^rf-'-z*'' ; 

\+x   '  ^^^  '  ^*'^  '  ^^-^     x{a*-x')    '     x(a'-x')' 

T(a+x)  {x-yY  xix-y)  a;'  (\+x^)(\-x^) 

x{a*-x*) '        ^•'•^  {x-yy  '         {x-y)^  '         {x-y)' '  ^     ^^  {l+xV  {i-x') ' 


2(0.7;' -16)  fl.r«(ar-4)  3T(2+3i,)  ^^    a+6. 

ec  ^*-^  ar(9j;«-l(>)'        ik(9j;^-l«)'        32(9j;'-16)  *  ^      '  abc  ' 

10a*  X*  adfh—hcfh         m'^+mn*—m*n  afd+ae 


20b*y^+7c*x*  '      9  +  Sxy^  '       bdeh  +  bdfg'  m-ii  '       bdf+be+cf 

,,.,  886a  6(x-7)  2  t5x  +  l  . 

<l*)li55'  —49-'  nT-         «~^^  i^;  ^^ 

ar«-f2ar4-13     ^       ,^i. .  2(a«+a;')  ^    2rt!»a;-3c.r2      ,^^^Sb'-2aV)-2a-S 

-V^T27- ''  ^'    <*^-)  -^^ ^  ^-^^-^ &r--  •  ^^^-^ a(6^^--rT)  —  • 

2  3a;=»+20x'-32g-235       ^ .  2«+(7+e  y-px-'^mpy* 

a:*+a;*-fl'        a;^+8a;*-5x-84     *     ^      ^rt  +  c)(a+d)(«+ c)  '       (3wy-'-.r)^     ' 

?±.^         „8.)0.         (19.)^,=      4,      ^-^^-^,.         CiOO^.^^i; 

4fl&  r21  ^         2^-3 ^oo  \  ?^  .     ^'  +  ^'  .      1  .    a''-(vc+ny-xy 

(a_6)«*        ^      ■'(x«-l)(2x+3)'        ^—^3'         a      '        '    b^-bx+by-z-y' 

a■>c4■&^c-2c"-^»  1        1.2         1_1.     1  .;^*_i3;s_a..5. 

(,^A«-«)ft?(«-/<))(2a'«_36)'        '       i*'     a2"*"ac'^  c*      &' '  '         ''"    ' 

91a«  1  Ta^i^  1  J}_ 

^^'•^  "66^ '        ^;r^  '      ^  '     *•  ^^^-^  121^ '  1^81^""  '  a:'^'*"  * 


ANSWERS.  307 

(29.)  !  +  ..'  +  «-       3(a  +  J);      ^Z^;      -J_; bcic-W_ 

-mhi-^^mn-^^n-^;         -^f^-y         a^-2ac  +  c^-b^ ;        1.  (31.)  1. 

(32.)  1.      (33.)  a-^-a-^b-^+b-^-a-^c-^-b-^c-^  +  c-^.      (34  )  ^'^'^' +^''''^'  . 

,  /m—n       (wi  + ;?)« 


INVOLUTION. 

(Page  59,  J»0.) 

,^  X  «  «     .  ?  ,         9         9      ,25      1    7»* 

(1.)  9a6;  4a«a;2;  ^^-;j7^ ;  ly^*'  ;  J-^;  g^  "f-  (2.)  l-2a;  +  3a;«-2a5»  +  a5*; 

4a2-12^za;i  +  9x'5.  (3.)  9 -  1.2a; -2.^2  ^ 4c' +  a?*;  27a;6 -27a;* +9x^-1 ;  l-2a;^ 

+  ar;  a;^— 3x^^^-3a;-y-y^        (4.)  Sla^a;^ ;   IGaV^ ;    a-^"'x-'^',    aW  ;    a-a;^'; 

125a;  V^*  «"'^"        (1681)  V 

T^'  -^r-        (<J-)  a;^+7a;6y+21x»y2^35a!V+35a;V+31a;V+7a'y6+2/^ 

a;*-42;»y-f-6a;«y«-4a'y»+y* ,    27rt'5-27a*a;+9a'^a;2-a;=* ;   a!-«-5a;-«2^+15.r-V'^  — 
35a;-8y='  +  70a;-»y4  -  etc. ;      X'^  +  4a;-5y  +  lO.^;-^^*  +  20.a;-''y='  +  35a-8^*  +  etc.  ; 

A        ^*  a;*  a;^  5a;»  1      ^  ,  3«^  ,  ^^^1       ?:'>^ 

'■'2"T5i      8":?^;^~l"^r^^^^      i.^2.^+8.-'' ^10.' +128.3 

+  etc.;        l-4.^  +  6.-4.c^.«;       -*;       l^{l^^^^^^^^ 


4-  etc.  )  •    *  *  *  •    *  *  *^        (7  \  4f  *  *  *  *^ 


(Page  61,  W.'j,) 

(1.)  2  -  2  -  3  .  59  =  472 ;  2  ■  73=146 ;  5  •  7  •  67=2345  ;  *  *  *;  ±  {a  -  c  •  (»+&)} 
=  ±(«'c+a/>c) ;  *  *.        (3  to  6.)  *  *  *  * 


308 


ANSWEIIS. 


(Page  Co,  J97.) 


To  give  the  roots  in  problems  in  evolution  would  be  to  destroy  the  benefit  of 
the  exercise ;  hence  they  are  omitted. 


REDUCTION  OF  RADICALS, 

(Page  70,  20^,) 
(1.)  *•*».        (2.)  *;  ^Vl8;   *;  *;   *;    Vl ; —{a^-h^)^  ;    **»***. 


a—b 


— rT—^— T —  Vl^x*—y*).        In  such  examples  be  careful  to  leave  only  integral 
forms  under  the  radical  sign,  in  the  reduced  expression.        (3  and  4.)  *  *  *  *, 

V\  =  V\.  (5.)    *  *  *,  a'A  -^y=ia*-b*)^.  (6.)    V^  and    Vd  ; 


2«  4/T«  .  •  *  ♦  •  .  ^"'^' 


»*»♦.    4/^:^11^  and  V7+]^.  (7.)   *♦*.         (S.)  ^VTS;**** 

o  if 

aj-y        '  2  .    ' 


g«+a;    a'  +  a;«^^  2a;«  +  l-2^  i^a;«  +  l ;  -(«+ VS^^l)  ;  a;*+a;+ VJ^+W+aj«--l; 
a* 

2(_  V6+  V2  +  2)  ;     2»^3^i^30-.^>^2  .  (lo.)  a^V^/Zy^+^^/^/ay 

+  j;  =*  2/'^  +  .r  'y  +  x  'y^  +  x^y^  +  .c*y'  +  .-c^y'  +  ar'y*  +  ^y ^  +  ar 'y  *  +  a;^y  "  +  y  '^ ;    ,r  ^ 

ia    1  1  14    ^  11  Ji  IXt    g  1   21  .    ^  4    21  2    15. 

+  X  ^  y*  4-  aj^'y*  +  .t  '  y*  +  a;  *  y '  +  a;^y  *  +  a;  'y*  +  x^i/  *  +  x^y^  +  x-^y  *  +  .x'y  * 
+y";  (i/8+  V^-  Vs)  (3-24/c).  (11  to  18.)  »  *  * 


COMBINATION  OF  RADICALS. 

(Page  74,  ;^77.) 

(1.)  *  *  *.        (2.)  *  *  *;       ^  ;        — .        (3,  4,  5.)  *  *  *.        (6.)  4^864; 

X  X 

fi 19, 10  f, 

iV  ^151875;       .V^'x'';  QxVxy' -,        Vl+5a;«  +  l(te*  +  10a;«+5a.'  +  2;W 


ANSWERS.  309 

i'i'W^:^;  dVWZ;        30;        12  Vi.  (7.)  41 ;      x+y;     246+581^ 

-11/^-36  i^d;        3'1^30-12V'3  -  V180  +  12.  {S.)lViO;       AVd; 

Vdi^;      IV'IO.         {d.)\^~i-kV2;    2xi^x;   iVQ  +  V2;       b  Vx  ;        f-' 


(a-/*)  (a -hi) 


^V4«*x;     ^A.Vl5;    5-2  VQ;    27{a-x)Va-x;     J-3ab^+dah-b^. 

(11.)  9^  V5P  ;      ia;2  VJ  ;         by  5  \^y  ;      2 j/7a;  V'sT ;      Vl^x;    i  1^3125^ ; 
i^3;        (12.)  2+31/5";        2V3+34^;         ^V^-i^^;        ***. 


IMAGINARY  QUANTITIES. 

(Page  78,  ;^;?,!/.) 

(2.)  ISC^^+l)^^^;     19a  V^;    (463+3c)|/rT.       (4.)  (4^±  V"^)  ^^. 
(5.)  1  V^  ;  12(  V3~-  1)  V~^ ;  11a  V^  ;  {a  Vb  -  c  Vd)  V^. 

(6.)1^^-        (7.)  5^2  1^^;    IQV^V^',    \'V^',     P7ii^^. 

(Page  79,  225.) 
5-7  V^,  and  9  V^-1 ;     2a+(  Vb+  V7)  i^^,  and  (  VT-  V7)  i^^. 


MULTIPLICATION  ANB  INVOLUTION  OF 
IM AGIN  ARIES. 

(Page  80,  226.) 

(2.)  *,   *,    -6>v/6.     (7.)  *,    3,    V^.     (9.)  278'v/^,    or   278^3  V"^ 
972 V^,  or  972v'2  y/^,    (10.)  icy^M'. 


DIVISION  OF  IMAGINARIES. 

(Page  81,  227.) 

(3.)  ^iV4  4/~l;      -i^aV-l.  («•>  -TI3)^- 


310  ANSWEllS. 


PART   IL 


SIMPLE   EQUATIONS. 

(Page  87,  28.) 

3&  b        a+h+c 

(l.)12;     24;     23i;      3.S;     8;     4; -;     -;      —7^;     2.9;    2|f 

^Q>>   J      6c-&*_      ^.      n(9'-;))       5rt(26-«)        8«6*+46»-12a2i  2k« 


a  m  "6c— d  Sa'^-hab—ac+bc  '     2Sp+6g'^ 

4;     4i  ( 

2«Vli  81 


^p^^ ;    8;    0;    3.  (Ji)  --,      4;     4;     4^  (4.)  81 


c«-2/>c;      16;      5;      4(a-l);     8;1;6;6;3;      ± 

6+1  a 

\       a       )  \        \ft  +  l/    j  '  (w+l)«'  26^        ^  ' 

r  ^      \    36    /   '  2a     '         16 '  4  • 

APPLICATIONS  OF  SIMPLE  EQUATIONS. 
(Page  90,  S3.) 

(1.)  ^'8  84,    J5's  42,    (Ts  14.  (2.)  ^'s^— ^^-,        5's        ""* 


1  +  n  +  mn'  1  +  n  +  mn' 

(7.)  :rr^-.   (8.)  i    (9.)?-"^|?.     (18.)  m  of  an  hour ;  '""* 


mn—m—ns  ^      an  +  bn  ns  +  ms+mn 

(14.)  317.951,1268,2219;  ^V-    ?^  •    f-T™^    ^'--         ^S-)  90; 
^      '  3  +  4m       3  +  4w        3+4w         3  +  4m  ^ 


g  +  6  .- ^ V  ^^^    mpa—pa  +  mb—na  +  mc  ^       ^ 


b'—ac 


p  +  2—m—n'                         '           m  +  n—mp+p  '               '      '     a—2b  +  c' 

(18.)  19, 30;        2±^,^,    2!!t^i.  (19.)  73,  77;       ^=^  , 

^      '                         m  +  1               7W  +  1  2m 

2m  ^  m  +  n      m  +  n 

33i  50;     14f,  28i,  42?.              (26.)  1200.  (27.)  50.              (28.)  5712. 
(30.)  50. 


ANSWERS.  311 

SIMPLE  EQUATIONS   WITH   TWO   UNKNOWN 
QUANTITIES. 

(Page  96,  4=2.) 

(1.)  aj=10,y=3.  (2.)19,2.t  (3.)  16,  35.  (4.)  7,  2.  (5.)  7. 17. 

(6.)  2,  2.         (7.)  12,9,  (8.)  -2,  19.  (9.)  -2,  1.         (10.)  ^J,    ^. 

(„.)i,  i       (12,)?^^+^,   2^^!+!'.      (13.)  ,-21,   ''1-^. 

ab      cd  da  *  '6b  ^      '  b+c  a 

(14.)  (a+by,  {a-h)\  (15.)  10,  5.  (16.)  18, 12.  (17.)  i ,     -  . 

a       0 

^^^•^%'     WV        ^^^■'^\'     li-  (20.)  20,  5;    6.8;    7.10;     8^^-72=0; 

y«-22y+120=0;    y^-Ay^  ^  14^4  _. 203,2 +9=0. 

APPLICATIONS. 
(Page  98,  42,) 
(1.)  18f ,  31^.        (2.)  3.        (3.)  20,  8.        (4.)  5000,  5000.        (5.)  ^.        (6.)  24. 

(7.)  29,  32.  (8.)  5000.0.  pm^gn-qmn  pmn-gn-pm  ^ 

mn—m—n  mn—m—n 

(10.)  ^±^-,     ^t?.        (IX.)  48,  16.        (12.)  24,  32. 
mn—\       mn—1  \      /      »  \      1      > 


SIMPLE  EQUATIONS    WITH  MORE  THAN  TWO 
UNKNOWN  QUANTITIES. 

(Page  101,  43.) 

(2.)  4,  3,  2.       (3.)  2,  3,  4.        (4.)  24,  60,  120.       (5.)  64,  72,  84.       (6.)  3,  2, 1. 

(7.)  2.),  5.,.  65.  (8.)         26c— ^  2ac        '  2a6        '  ^^^ -Q  ' 

-h     W  ^l^-^l'     I'    ^'         (n.)2a,2b,2c.         (12.)^^!^, 

1  1  (13.) -f-^.   -^.    -M.        (14.)  12,  5, 


{b—a){b—c)'    {c—a){c—b)'        ^     ''      6-f-c'        a+c'        a+&* 

7.  4.        (15.)  2,  1,  3,  -1,  -2.        (16.)  3,  4.  5,  1.  2.        (17.)  h  }-c-a,     a-hc-b, 

a  +  b—c. 


t  The  valacB  of  the  unknown  quantities  are  given  in  tlie  order  x,  y,  Zy  etc. 


312  ANSWEUS. 

APPLICATIONS. 

(Page  102.) 

(2.)  |2,  20  cents,  10  cents.   (3.)  £3000  at  4%,  etc.    (4.)  ^  ,  ^,    ^, 

o7    ot         o7 

2Sa 

— .    (5.)  142857.    (6.)  26,  9.  o.    (7.)  140,  00,  45,  80.    (8.)  18,«^j,  34f, 

23/i ,  80. 


BATIO, 

(Page  105,  50.) 

"•'^=^  1=  7=--r„^  S—      <■■  ^^ 

4 
3  ' 

3 
5a' 

.W-      9=     Z'     .i*-     (*.)5:n;l:«'+6.;2(a    x):(««). 

(4.)  9 

:25; 

a«:6«;      27:125;      a^zft^      5:4;      V^iVl;       Vm.Vn;      3:4; 

^7: 

^-y. 

(5.)  The  former.        (7.)  4:1. 

rROPORTION.—APPUCATlONS. 
(Page  111.) 

(8.)  13,26.  39.       (4.)  8,0.       (5.)  ^^  +  |-^ ,    2^-|i.       (G.)  120, 160,  200. 

(7.)  8:9.  (8.)  252.  (9.)  56,84,  70.  (10.)  20.  (11.)  150.  (12.)  300. 
(14.)  3h.  49j\m.,  3h.  32i\m.,  •  3h.  lOAm.  (15.)  Every  h\  hours,  A  hours, 
and  1  ,\  ;  or  11  times  in  12h.,  22  times,  and  11  times.  (1(».)  No  ;  since  it  talces 
the  minute  hand   1-^-  hours   to  gain  a  round,  and  ^  to  gain  half  a  round. 

(17.)  8:45  A.M.        (18.)  1st.  ^-^.     -^.  etc.;     2d.    -i=-^,     |^-^,  etc.; 
M—m      M—m  m—M       m—M 

a+mt        s+a  +  mt        2s+a+int  s—n—mt        2s—a—mt 

9—a+Mt      2s—a-\-Ml 
M—m    '        M—m 


etc. 


ABITHMETICAL  BROGBESSION. 

(Page  117,  8:i,) 

(1.)  83,  903.        (2.)  -39,  -384.        (3.)  ^-^  ,    ^^^ .        (4.)  0,  "^ 


ANSWERS.  313 

(5.)  193     243  .  •  293  •  343  •  •  393  •  •  443.        (7.)  -46,  | .     (8.)  100.      (9.)  ^^^, 
tn+n      m+Sn 


GEOMETRICAL  PROGRESSION, 
(Page  120,  90.) 
(1.)  46875,  58593.      (2.)  6, 18,  54, 162,  486.      (3.)  16384,  21845^.      (4.)  —h, 
-hh       (5.)il.n,§i        (6.)¥;    .3;    ^g;    If.        (10.) -A[(-^3)»-iJ ; 
nm;   I;   Wo\%^;   i 


VARIATION. 

(Paoe  124,  95.) 
(6.)  «az.        (12.)  18.        (13.)  «=i/i*. 


HARMONIC  PROPORTION  AND  PROGRESSION. 
(Page  126,  100.) 

(6.)i,TH-,T»J. 


PURE  QUADRATICS. 
(Page  128,  iO«.) 
(1)  ±4.        (2.)  ±5.        (3.)  ±  V2a^  -  &».        (4.)  ±  V6.         (5.)  ±i^»V3. 
(6.)  ±  6.      (7.)  ±  j/^.      (8.)  ±  V:^.       (9.)  ±  ^  V5.      (10.)  ±  3  V=^. 

ai.)  ^|V-io.     (12.)  ±f     (13.)  vffi^p.     (1^-)  -  -S;:;^""- 


APPLICATIONS. 


±  w  l^i 


(1.)  12, 20.  (2.)  ±  ia  VS,  (3.)  -^^====^^^ .        V^nJT^^' ' 

±pV7  /4X4K^  /K^    ■ - —      _^^-IL-.  (6.)  5.57+, 

18,12_,  40.51  + .    (7.)  149,247.2  +  miles  from  the  surface  of  the  earth     (8.)  240. 
132  VL  from  A.        (10.)  -^  from  the  louder  bell. 


314  ANSWERS. 


AFFECTED   QUADRATICS. 

(Page  133,  114.) 

(1.)  8,  -2.        (2.)  6,  2.        (8.)  o(2±  Vn).       (4.)  8,  -1.        (5.)  2  ±  V^^. 

(6.)1. -a.  (7.)  2.-1.  (8.)7,i  (9.)-,-.       (11.)  3, -6g. 

(12.)  5.  -^h        (13.)  ^^ .         (14.)  t,  A.        (16.)  Wm,  6.         (16.)  2,  -?. 

(17.)  ~.    ^ .         (18.)  i(-l  ±  i^l33).        (19.)  4.  i.         (20.)  ia(-3  ±  V-?)- 

(21.)  ±  k  ^^.         (22.)  8,  -%.  (23.)  ^^^ .  &.  (24.)  4,  -i 

(25.)  12,  4.  (26.)  4,  i         (27.)  7.12+,  -5.73+.  (28.)  ia(l±3  V^). 

(29.)  1,  gV.        (80.)  5,  3. 


HIGHER  EQUATIONS  SOLVED  AS  QUADRATICS. 

(Page  136,  122.) 
(1.)  ±  3,    ±3  V^.       (2.)  2  ;  the  other  four  roots  not  required.       (3.)  ±  m^. 
(4.)  27.        (5.)  121.        (6.)  64.        (7.)  b^.        (8.)  ±  8.        (9.)  ±  V^,  ±  V2. 
(10.)  V'K-I  ±  Vl^i).        (11.)  4,  M  (12.)  I  ^  (-^  =t  V4^+T«)  [  ^• 

(18.)  V'C/z  ±  V}^^T^\  (14.)  243,    (-28)1  (15.)  16,    ^^^^i. 

(16.)  |^(J±  i/ft^Ti^i;;)}  '.  (17.)  8,^.  (18.)  ^,|/I^.  (19.)  1. 
1,  1  ±  2 1^15.  (20.)  0,  -12.  (21.)  3.  -i,  i(5  ±  Vl329).        (22.)  4,  69. 

(28.)  i(l  ±  V5).       (24.)  |(l  ±  i/5).        (25.)  3,  h  U-8  ±  i^55).       (26.)  2,  H. 

i(7  ±  Vm.  (27.)  |7^^-  (28-)  5,  -1,  2  ±  i^^H:  (29.)  5.  -2, 
i(3  ±  /=15).  (30.)  2,  3,  1.  (31.)  2.  -|,  i(l  ±  V^Z).  (32.)  1,  -3,  -3. 
•(33.)  2,  2  ±  V^.  (34.)  5,  4  ±  V^  (35.)  -2,  -1,  -5.  (36.)  6,  20,  3. 
(37.)  6,  4,  5.  (38.)  1, 1,  -2,  -2.  (39.)  4,  1,  3,  2.  (40.)  3,  -1, 1  ±  i^^. 
(41.)  5,  -4,  3,  -2.  (42.)  3,  -3,  i(-13  ±  V^ISS).  (43.)  4,  3,  i(7±  V69). 
(44.)  9, 4.  i(-3±  V^l  (45.)  + 1,  -1,  ±  V^^i: ;      -1,  Kl  ±  ^^)  ;     ^ 


±1, 


v^. 


ANSWERS. 

V2 


315 

i(±  Va  ±  i^2  ±  V-2  ±  V2). 


(46.) 


2a±V2{l 


2(1 


;?(1±^)  ±   A/(^^^^a+<^)Y_r  (48.)   i(a  ±  V7^-4\ 

-a)  ^    \       2(l-«)       J 

(49.)^±  iV-3±i^8(^).         (50.)  i(l  ±  VE).         (51.)  2,  -K  i(3  ±  I^SOS)- 
(52.)  1,  1,  -2,  -2.     (53.) I  (-1  ±  V~^).       (54.) ±  -^  i  -^^^^  +  V^6t^T2  [  , 


V  a  (       ^  4  , .  \    , 


(55.)  ±-|(Vl  +  a8-l)(Vl-a2  +  l)'f^  4,i. 


(56.)  0,i,f     -1,  +2,  -2.       (57.)  h  i(-l±  ^-35),     ±1,  ±Vi(-ll±v«5). 


SIMULTAJSTEOUS  QUADRATICS. 

(Page  142,  J;37.) 

(1.)  x=%,  -I? ;    y=-4,  W.  (2.)  a;=±  t'f  ;    ?/=2t  V  i         (3.)  a!=2  ; 

y=2.        (4.)  «=±7,    ±4;    y=±4,±7.  (5.)  a?=±3,  ±11^2;    y=±2,±\V2, 

(6.)  a;=±2,  ±i  4^10 ;      y=±i,   =Ff  i^.  (7.)  .x-±3,     t8  ;  2/=±5. 

(8.)  a;=±f  i^li;  y^ilVTi.  (9.)a;=±f4^;  y=±|4/2i.  (10.)  a-=±l. 
±V-4^^;  y=±2,  +^4/^=5.  (ll.)a;=±2,  i^i^jT;  y=±6,  ±  ^i^  i^. 
(12.)  a;  =  ±  10,    ±  H  4^"=^  ;       y=  ±  3,      =F  ^  4^^=47.  (13.)   aj  =  4,    2, 

i(-13  ±  V377);  2^  =  2,  4,  i(-13  T  I^^TT).  (I4.)a;=4,  -2,0;  y=2,  -4,  0. 
(15.)a;  =  2,3;  y  =  3,  2.  (16.)  a;  =  ±  3  ^2  ;  y  =  ±  V2.  (17.)a;  =  9,  4; 
y  =  4,9.  (18.)  a;  =  11,    i(l  ±  V^^l)  ;         y  =  3,    i(- 15  ±  V=^l). 

(19.)  a;  =  15,  0 ;  y=45,  0.        (20.)a;=±V2;    y  =  2^V2.  (21.)  a;  =  0,2; 

^  =  -2,0.  (22.)  a;  =  1,4;  y  =  4,  1.  (23.)  ar  =  1,  3,  2  T  5  f'^;  y  =  3. 
1,2±5V^.  (24.)  a;  =  5,  -2,  K3±V^^67);  y  =  2,  -5,  K-3  ±  V'^^)' 
(25.)  a;=±3,  ±2;    y=±2.  ±3.        (26.)  x=  ±2,  ±1,  T  2  4^^,  ±  V^  ; 

y=±l,  ±2,    ±4/31,  ^2^^=!.         (27.)a;  =  \^K^2-l);    y=  ^2(V2-^' 

2abc  2€ihc  2ahc 

(28.)  X  = 


y  = 


7 7,      y--j— 7 ,       ---r T--  (29.)a;=±3, 

ac+bc—ab  ab+bc—ac  ab+ac—bc 

y=±2,  s  =  ±  1.  (30.)  a;  =  ±  2,  y  =  ±  4,  2  =  ±  6.  (31.)  a;  =  1,  2/  =  2 
2  =  3.  (32.)a;=±V,  T  %^  ^^,  ±5,  ^4^/^;  y  =  ±  ^sS  ±  ¥  ^^ 
±  4,    ±  5  i^^.  (33.)  a;  =  ±  f  1^2,    ±3,    ±3  V^  ;        y  =  ±  '1^2,     ±  1, 

±  /Zn:.        (34.)  a;=8,0;y=8,0.   (35.)  a;=2,  8  ;  y=8,  2.        (36.)  x=10T4.Vn, 

(37.)  a;=  ±||/±15,    ±^V±3a: 


*0  T  I  Vl5  ;    y  =  10  ±  4  V6,  1*0  ±  |  VI5. 


316  ANSWERS. 

±aVWl',     y=:±yV±3,0.        (38.)  ^  =  4,  9,   -3T44^iO;     y  =  9,    4, 

-3±4iC:iO.        (39.)  a?  =  2^(15  ±6  1^^,5,1;   y  =  M^S  ±  10  V^\  3,  I 
(40.)  «=  f  ;    y=16.       (41.)  a;=4, 1.0;  y=8,0.     (42.)  25=2744,  8  ;    ?/=9r>04,  1. 

APPLICATIONS. 
(1.)  3.        (2.)  18,  $20.         (3.)  10  and  3  days,  120  and  36  miles.        (4.)  12,  CC. 

(5.)  14.  10.       (6.)  6  miles  an  hour.       (7.)  4  and  5.       (8.)  J  ^(^^-l)P_^jy\ 

\  Vm—1         / 

VVj^P^Tj^Y        ,9)j^.^(5^^)         (10.)  1,3,5,7.  (11.)  2, 
\           Vm—1          / 

3,  4,  5,  6.        (12.)  3,  6,  12.        (13.)  2,  4,  8.        (14.)  5, 10,  20.  40.  (15.)  2,  4, 

8.       (16.)  6,  8,  10, 12.       (17.)  1,  2,  4,  8.       (18.)  108,  144,  192,  256.  (19.)  72, 

63,  66.           (20.)  7,  3.           (21.)  25.            (22.)  $960,  $1120.  (23.)  248. 
(24.)  6  and  7  per  cent.        (26.)  3  and  14. 


IJS^EQ  UA  LI  TIES, 
(Page  150,  134.) 
(8.)  H  and  V-        (9.)  Any  number  between  15  and  20. 


PART  III. 


DIFFERENTIA  TION. 
(Page  157,  156,) 
(3.)    l^bx^dx  -  mxdx  -H  4rfj-.  (4  )    2Axdx  +  ZBx^dx  +  ^Cx'dx. 

+Qdx;  (5.7;*-12.c='+12a;*-2a;)rf«;  {x-2x^  +  l)dx.      ilSton.)15{a^+x')'x'dx ; 

^{Zx-2)^dx ;        i{2-x'y^xdx ; -—.  .  (18  to  22.)  -  -r-^  . 

2(1 +a;)^  ^^"^^'^    ' 

—  t:, T^i :     —  75 rr;     ts rs  .*    Ti a  •         (23.)  When  x  >  1,  faster;  also 

{1+xy  (!+«)*      (l+ar)«  '     (l+x)* 

faster  when  x<i  and  > .     When  x  = =  ,  they  both  change  at  the  same 

dV2  SV2 

nte.     When  x  < ,   y  changes  slower  than -a;. 

3^2 


ANSWERS.  317 


INDETERMINATE  COEFFICIENTS.— DEVEhOPMENT  OF 

FUNCTIONS. 

(Page  161,  161,) 

(3.)  l-ia;«-ia5*-etc. ;     x-^^+x^-x^+etc, ;      ^  +  ^x+^x^  +  ^x^ 

-»-etc. ;      l4-ia;+t«2_|_.i.a.3^.;^4^et(,^  (4.)  S+Saj-ac^-Saj^'-aj^-etc; 

l  +  2x  +  a.»  +  te'+9.V4-etc.;  ^  -  |^  +  ^  _  g,  +  ^,._et. 

(5.)  1— 4iC— ^a;2— 8^5^— etc.;     l+^x—^^+-^^x^—eic. 

DECOMPOSITION  OF  FRACTIONS. 
(Page  164,  167.) 
5  3  3  15  4  7 


(2  to  6.) 


3(a;-2)      3(a;+l)  '         2(a;-2)      2a;  '         a;-4      a;-3 '         2(a;-4) 


_       1      ^    _J ^  9  _2 1__      _3_ 

2(aj-2) '  2(a;+l)  a;+2"^  2(a;+3)  *  ^  **  ^^  (aj-l)^  (a;-l)=^  "^  a;-l ' 
2 3__        _1_  .      1  _  J^        2        _J 1 !___ 

{x+ZY  (a?+3)*  "^  aj+3'  a;*  jc*  +  3.  +  /^\-x)  2{l+x)'^  4(l+a!)' 
1 1 \__  ,       1 2__      1__      2__ 

4(a;-l)      4(a;+l)      2(a;«+l)'       25(a;-2)2       125(aj-2)  "^  25(a5+3)2  "^  125(a;+3)* 

ft^       1  2^-2         !       3(a;  +  4)  1  1         J^  _  2      _3_ 

(12  to  18.)  ^e:^^      (a;2+l)2'     a;  "^  (aj'^-2)=^  "^  a!2-2  "^  a;-l  '    a;=^      a;  "^  a;+l ' 

1  (_2_    _    _1_  g-2       _       a;+2     j  1  1__ 

elaj-l  a;+l    "^   a;*-a!+l  x'+x+l)  '  Aa\a+x)        4ta\a-x) 

1_ 1 !____.    J.? ? 12_ 

■^2a«(a*+a;«)'    (a-6)(a;-a)      (a-6)(a;-6)'    a;-3      a;-l      a;-2* 


tjtje:  binomial  formula. 

(Page  167,  i7J.) 

(1  to  6.)  a^-Qa'h-'rl^a^h^-'ZOcL'h'-^X^a^h^-Qab'+h^  ;       a;^-7a;6y+21a;«2^« 

-  35ajV*  +  35a;'y*  -  2\x^y'  +  Ixy^  -  y' ;  a»  -  na**"*  aj  +  ^^^^—  ^"~'  «^ 

^  M^-l)(^-g)^n-3^.  ^^(^-l)(^-2)(^zi)^.-4^4_etc.;    l4-4.+6.^-f4.^ 
^  if 

W^_l)  w(7l-l)(7l-2) 

+«*;      1-5^4- 10y2_l0yJ+5y4-2^«  ;       l-ny+  —3 — 2/' jg y' 

_^^^-l)(n    2)(7^-j)^^_^^^^      (7  to  11.)*****.     (13,  14  to  17.)**** 

fl8to20.)  a^-\a  H^-^a    ^x'-ha    ^a;6-etc.;  ^-5  +  ^^  +  gj-^ 


818  ANSWERS. 

i(V|.6  i  a  _3, 

+  i^29^  +  etc. ;         a'+4a^c''+6a*c+'ia'c'-\-c\  (28.)  -^Siia  ^  b'\ 


LOGARITHMS, 

(Page  179,  199.) 

(1.)  4,  6,  *  *  *  (2.)  -2,  *  *  *.  (3.)  4,  *  *.  (5.)  To  the  3291147th 

power,  and  the  1000000th  root  extracted.  (G.)  The  1000000th  root  of  the 

3414639th  power,  (7  to  9.)  *  *  *  *.  (10.)  .23108,  .17677,  *  *.  (11.)  4.449419, 
4.637084,1.890210.  (12.)  12.42,  .00010031,  18.3625,  1.8358.  (14.)  flog  aj 
4-i[log  (l+ir)+log  (l-«)],  Klog  rt+log  .^■-  log  6-log  y),  ^[log  («-«)+log  {s-b) 

+log  («— c)— log  *],  ^[log  ar+log  (1— .r)]  — J  log  y;      -  {m  log  a+p  log  6- Hog  c), 

i  log  c—  -  log  (f-H  -  [log(7/t-har)+log  (w— .t)]— w  log  a+n  log  6.        (16.)  —  z 

r;         a;  x     2{l  +  ir) 


SUCCESSIVE  DIFFERENTIATION, 
(Page  182,  204,) 
(2.)  12a;da;«.        (6.)  *  *  ♦.        (7.)  2[{x-h)-\-{x-c)+(z-a)]dx*. 


DIFFERENTIAL  COEFFICIENTS, 

(Page  185,  207.) 
(6.)  10**4-12aJ*-l(te,  40«»-f36x«~10,  120x«+72ar,  2405+72,  240;    ****. 


TAYLOR'S  FORMULA. 
(Page  188,  J^i;?.) 

4-2aJ-»y+3ar-V*+4ar»y3^.5a^6y4^({,,-7y5^_e^c.  ;  a;~^-p"^y+t«"V 

_|^^-V'y3+^a;*'^2/*-^f«"^^'^^/'  +etc.     Page  189,  (2.)  3aj«-2aj2+(15aj4-4c)A 
+(30ic3-2)A2  +  SOar^A'  +  l&r^^  +  ZhK 


ANSWERS.  319 


INDETERMINATE  EQUATIONS. 

(Page  192,  218.) 

Sy=    5,    14,   23,   32,   41,   50,   59,   68,   77,86,95,104,113,122,131,140,149. 

U=215, 202, 189, 176, 163, 150, 137, 124,  111,  98, 85,  72,  59,  46,  33,  20,  7. 
.      Sy=2^,    Q.     ,^.    (y=  8.  (2^=  9,    28.    47,  etc.  (y =2, 119, 236,  etc. 

^      i  x=Vll,  28.     ^  '    ( a;=20.     ^-^  ^  { x=ZQ,  173,  290,  etc.     ^^^  \  a;=3,  131,  259,  etc. 

i^)\ll\  »5.+9.=40.       None.        5.-9.=40,  |fl,?;  ^J  J -• 

APPLICATIONS. 
(2.)  Yes ;  15, 163,  9.        (4.)  No ;  yes,  in  an  infinite  number  of  ways ;  4  3-shil- 
Img  pieces  and  192  guineas ;  possible  ;  possible  ;  possible.        (5.)  190. 


INDETEBMINATE  EQUATIONS  BETWEEN  THREE 
QUANTITIES. 

(Page  194,  219,) 

(z=  1,    2,    3,    4,    5,    0,  11,  12,  13,  14. 
(2.)]y=ll,    9,    7,    5,    3,    1,    8,    6,    4,    2.  (3.)  59  sets  of  values. 

(a;=10,  11,  12,  13,  14,  15,    1,    2,    3.    4. 

(A)z-l\^=^'    4' ^'8' 10-^    2-2^^^='^'    3,5,7,9.)  (y=  2,4,6,8. 

^*-^  '-^  U=15,12,9,6,    3.)  <.r=rl4,  11,  8,  5,  2.  f  U-=; 

(a;=r9,6,3.f  f;r=5,  2.1  ]a;=4,  l.f 


:10,7,4,  1. 


(Page  194,  220.) 
(1.)  «=7,y=2,a;=10.         (2.)  e=15,  30,  ^^=82,  40,  a; =15,  50.         (8.)  None. 

APPLICATIONS. 

(1.)  8  of  1st,  6  of  2d,  2  of  3d,  and  in  9  other  ways  ;  23  and  2,  10  and  5,  9  and 
8,  2  and  11.  (2.)  $4,  $2,  $7  ;  infinite  variety  of  prices.  (3.)  6,  3,  1,  16. 

(4.)  Nuniber  of  the  3d  kind  equals  twice  the  number  of  the  1st  kind,  plus  tlie 
number  of  the  2d  kind ;  1  of  1st,  6  of  2d,  8  of  3d  kind.  (5.)  40,  60,  24. 
(6.)  55,  10,  85  is  one  result  in  integers.  There  are  an  infinite  number  of  other 
ways.    (7.)  *  *.    (8.)  s  :=.  10,  y  =  1,  x  =  13. 


320  ANSWERS. 


LOCI  OF  EQUATIONS. 

A  LAROE  number  of  these  constructions  are  exhibited  in  the  text,  and  to  give 
taore  would  be  to  destroy  the  possibility  of  the  student's  deriving  any  benefit 
from  the  exercise. 


HIGHER  ^0 17^  TJOJV^S.— TRANSFORMATION. 

(Page  205,  228.) 
(2.)  Multiply  by  y*,  and  then  put  y=x^°.     Finally  put  x=  —  ,  etc. 
It  is  not  deemed  expedient  to  give  farther  explanations. 

(Page  214,  249.) 

(2  to  84.)  To  give  the  roots  of  these  equations  would  destroy  the  practical 
Talue  of  the  examples. 

(Page  216,  250.) 

{1.)  x^-2x*-Ux+12=0.  (2.)  x*-2x^-5x'+4x+Q=0.  (3.)***. 

(4.)  x^-x*-7x+15=0.  (5.)  »  *  *  (6.)  d0x'-nx'-l]x+6=0. 

(7  to  10.)  *****♦.        (11.)  x^-\0x''+dSx*-5Qx^-7Sx^+66x~\-S9=0. 


EQUATIONS   WITH  INCOMMENSURABLE  ROOTS. 

(Pages  216-247.) 

To  give  the  answers  to  these  examples  would  be  to  destroy  their  value  to  the 
student. 


CARDAN'S  rROCESS. 

(Page  251,  281.) 

(2.)  -1,  2, 2.        (3.)  2,  -1  ±  Vs.         (4.)  V'i  -  '^'2.     and     the     roots    of 
w^ -^.{i^-l^)x  +  {i^i-i^)\e.=  0,      which      are      -^{i^i-h) 

(0.)  1,  -2  ±  3  V^.  (7.)  *  *  *.  (8.)  2,  2  ±  V^.  (9.)  8.  -4,  -4. 

(10.)  *  *  *.•         (11.)  *  *  *.  (12.)  *  *  *.  (13.)  One  root  is  2.32748+ 


AN8WEKS.  321 


VESCABTES'S  FMOCESS. 

(Page  252,  283.) 
Ex.  4,  -2,  -1  +  V^,  and  -1  -  V^. 


MECUBBING  EQUATIONS. 
(Page  255,  291.) 

(1.)  2  ±  V3,  ^  (1  ±  V^).        (2.)  -  1,  ^  (9  ±  V77),  ^  (3  ±  V5).         (3.)  1,  3, 

l+4a±V5  +  20a 


i  —  2,  —  f  (4.)  —  1,  ^m  ±  V|^m2  —  1,  in  which  m  =        „  ..       . 

^  (.1— «) 

(5.)  -1,  1,  I,  -1,  -1,  1  (1  ±  V:^).  (6.)  2,  ^,  ^m  ±  V^m^  -  1,  in  which 

m  =  1  (-  5  ±  V5).      (7.)  2,  ^,  2,  i,  ^  (1  ±  V-"3.      (8.)  |(3  ±  VS),  |  (-7±3  VS). 

(9.)  i(V5-  1  ±  V-  10-2  \/5),  _  ^  (V5  +  1  T  V-  10  +  2  V5).      (10.)  ^rti 
±  \/^rn^  _  1^  in  which  m  =  2  (1  ±  V3). 


BINOMIAL   EQUATIONS. 

(Page  255,  292.) 
(1  to  6.)  See  answe^'    to  (45),  page  138,  and  multiply  them  respectively  by 
\^5,       \^^,      fe,      ^,      V'n.  (7.)  See  as  above. 


EXPONENTIAL  EQUATIONS. 
(Page  256,  206.) 
(2  to  6.)  3.0957+,  11.384+,  3.292+,  0,  0.  (8.)  3.597+.  (9.)  2.316+ 

(10.)  2.879+.         (11.)  3.233+.  (12.)  2.001+.  (13.)  ^"^     i^g/'"^ 

/        2Jo|a+log_c^  lo|c  (ie.)2i,3|.       (17.)  (^)""^^ 

^      ^        Mogflj  ^      '  w  log  a+7i  log  6  \o/ 

(  jj      .  (18.)  ^[q  +  1^^;.    .|?  -  i^^j.  (19.)  3l^^_p3  i„g  3 

2(2+log5)_  243     _^  42^  J     I'^S-        (24.)  $196.71 

2  log  2+3  log  3  Mog(r?+6) 

$198.98,     $200.17,     $259.37,     $265.33,      $268.51,     $180.61,     $181.43,    $134.83 


322  ANSWERS. 

(25.)  7.13,  10.24,  16.23,  20.48  years.          (26.)  29.91  years.  (27.)  $1502.6a 

(28.)  11933.97.         (30.)  $1157.28.         (32.)  $4794.52,    $3500.  (33.)  $577.06- 

(35.)  The  former  by  $629.03.     (36.)  $500.91.      (38.)  13.58  years.  (39.)  $796.87. 
(40.)  $8229.70.        (41.)  Gains  $1756.60. 


APPENDIX. 


SJEBIES. 
(Page  275,  311.) 

(2.)  12, 6, 0.       (3.)  8,  32.        (4.)  -3^.       (5.)  1.        (6.)  -14. 

(Page  276,  313.) 

(4.)  -ar',   +««,  +x.      5x\  5x^.  (5.)  -27,  +9,  +3.      32805,  98415. 

m  +3x\+2x.    1093aJ^  3281a;«.       (7.) -4,  +4.    192,448.        (8.)  -2.7;*, +.r^ 

+2x.    87aj»,  173x^.       (9.)  -jx.     ^x\-^  x\      (10.)  -1.  +4,  -6,  +4. 

56,  84, 120.        (11.)  +1,  -3,  +3.    26,  34,  43. 

(Page  277,  314.) 
(1.)  4516a;'.      (2.)  17a;».       (3.)  -x\       (4.)  2733.       (5.)  29525.       (6.)  1365. 
(7.)  20.      '^1\         (8.)    n(n  +  l).  (9.)  6396a;"         (10.)  +1,  -5, 

+  10,  -10,  +5.     +1,  -3,  +3.     +1,  -3,  +3.     +3a;',  -x^ ,  +2x.  (11.)  »«. 

(12.)  8694.         (13.)  26.  34,  43, 53,  64.    26a;",  34a;",  43a;»,  53a;",  64a;".    196,  336, 
540,  8:5,  1210,  1716. 

(Page  279,  315.) 
(2.)  No.        (8  to  6.)  Yes. 

(Page  282,  316.) 


^'-f  l_3a;_2a;2  •         ^-''  l-x-x"^  '         ^^'^  {\-xY  '         ^    ''  l-2a;-a;«+2a 


5fctl) .  (10.)  S78256  ;    «°+10"'+35";'+50»'+jjg .  („.,  103155O  ; 

i  120 

3 .  (12.)  60710;        _+--+_-  -^. 


ANSWERS.  323 

KV^-^-^-^)-        (20.)  i        (21.)  i.        (22.)-A-.       (23.)  1. 
(25.)  ^i^,       (26.)  i.       (27.)  A-.        (28.)  -h       (29.)  f,.       (30.)  r^,-^. 

PILING  BALLS  AND  SHELLS. 

(Page  287,  322.) 

(i.)  1540,  13244,  903.    (2.)  9455,  4324,  35720,  465,  276,  1128.    (3.)  7490, 
3880.   (4.)  624.   (5.)  2730.   (6.)  36256. 

REVERSION  OF  SERIES. 
(Page  288,  ^323.) 

(2.)  a;  =  y  -  y«  +  y'  -  y*  +  etc.  (3.)  a;  =  y  -^  3y«  +  13y^  -  67^/*  +  etc. 

(4.)  a;  =  y  +  4y'  +  hy^  +  -j^sV''  +  etc.  (5.)  «  =  iy  -  Ay"''  +  f Ay*"  -  etc. 

(6.)  a;=(y-l)-i(y-l)2 4-^(y-l)-'-i(y-l)*  +  etc.         (7.)  x^^  +  {am^-r>)y^ 


[hm  *  —  mp —2in{am  * — n)']y  ^ 

+ .  —  +  etc. 


INTERPOLATION. 
(Page  290,  325.) 
(2.)  1.794.        (5.)  (♦  *  *). 


PEJRMUTATIONS. 

(Page  293,  334.) 

(1.)  720,  24,  3,628,800.     (2.)  720,  42,  210,  840,  2520,  5040,  5040.  120,  21, 

35,  35,  21,  7,  1.   (3.)  36,  84.  126,  126,  84,  36,  1.   (4.)  72,  504,  3024.  (5.)  20. 

(6.)  35,  21.     (7.)  127.     (8.)  479,001,600.     (9.)  792.  (10.)  15. 
:il.)166,320.  64,864,800.  (12.)  1023. 

PROBABILITIES. 
(Page  295,  335.) 
(1.)  U,  -h  ;    n,  4^.         (2.)  f,  f,  2:1,  4:3.         (4.)  6  to  1.         (6.)  (*  *  *> 

(6.)  i  5toi.     (11.)  mh 


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NOV  13  t93i 

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JUL  17  1936 


m. . 

Bity,  author 
ection  of  a 
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laterial,  its 
f  sentences, 
ormation  of 
criticism  of 


^♦-•1 


SEP  1ft  193? 
SEP   8    1941 

NOV    11  1942 
23Apt'64LM 
REC'D  LD 

APR  2  4 -64 -10  AM 


course.    By 
tiewisburg. 

or  advanced 


pages 

tvn   Univer- 


•wn  Univer- 
11  thousand. 


OCT  2  9  ^as^ 

It'  l'\ 


mU     APR     9  1982 


LD  21-50w-8,32 


Brown  Uni- 

it  of   Beloit 

leral  accept- 
used,  as  Dr. 
;he  tcanis  of 
many  years. 
id,  yet  clear 
rtant  branch 


i.7^  i^^^fc^^^.i^-^'A.i-*^ 


Sheldon  &  Company's  TextSooks. 

COLTON'S  NEW  GEOGKAPHIES. 

The  whole  subject  in  Two  Books. 

TJiese  hooks  are  the  most  simple^  the   most  practical,  and  best 
adapted  to  the  wants  of  the  school-room  of  any  yet  pvJblished. 

I,    Colton'8  New  Introductory  Geography, 

With  entirely  new  Maps  made  especially  for  this  book,  <  a 

II. 

\ 

and 

Elej 

Thii 
raphy 
and  er 
more, 
raphy 
School 

For 
we  ha"v 

Coltoi 
One  \  ^« 


800548        Oifi 


^ 


UNIVERSITY  OF  CAUFORNIA  LIBRARY 


A  very  valuable  book  and  fully  illustrated.  The  Maps  are 
compiled  with  the  greatest  care  by  Geo.  W,  Colton,  and  repre- 
sent the  most  remarkable  and  interesting  features  of  Physical 
Geography  clearly  to  the  eye. 

The  plan  of  Colton's  Geography  is  the  best  I  have  ever  Been.  It  meets  the 
exact  wants  of  our  Grammar  Schools.  The  Beview  is  unsurpassed  in  its 
tendency  to  make  thorough  and  reliable  scholars.  I  have  learned  more  Geog- 
raphy that  is  practical  and  Available  during  the  short  time  vpe  have  used  this 
work,  than  in  all  my  life  before;  including  ten  years  teaching  by  Mitchell's 
plan.— A.  B.  Heywood,  Prin.  Franklin  Gram.  School,  Lowell,  Zlass. 

So  well  satisfied  have  I  been  with  these  Geographies  that  I  adopted  them, 
and  have  procured  their  introduction  into  most  of  the  schools  in  this  county. 
Jajibs  W.  Thompson,  A.M.,  Prin.  of  Centreville  Academy,  Maryland. 


Any  of  the  above  sent  by  mail,  post-paid,  on  receipt  of  price. 


